Integer transform video compression system, method and computer program product

ABSTRACT

A video coding/decoding system, method and computer program product employ an integer transform matrix for transforming to/from transform coefficients and residual pixel data in moving pictures by a set of semi-orthonormal basis vectors. The basis vectors are derived from conventional DCT or KTL matrixes, but relaxes to some extent the requirements for orthogonality, norm equality and element size limitation. In this way improved coding efficiency and lower complexity compared to previously used integer transforms are possible.

CROSS-REFERENCE TO RELATED APPLICATION

The present application contains subject matter related to U.S. Utilityapplication Ser. No. 12/268,842, filed on Nov. 11, 2008, the entirecontents of which being incorporated herein by reference.

BACKGROUND

1. Field

This technology generally is related to video decompression andcompression systems, methods and computer program product, and inparticular to an integer transform function performed in those systems,methods and computer program product.

2. Description of the Related Art

The “background” description provided herein is for the purpose ofgenerally presenting the context of the disclosure. Insight provided bythe present inventor, to the extent it is described in this backgroundsection, as well as aspects of the description which may not otherwisequalify as prior art at the time of filing, are neither expressly orimpliedly admitted as prior art.

Transmission of moving pictures in real-time is employed in numerousapplications such as video conferencing, “net meetings”, television (TV)broadcasting and video telephony. However, representing moving picturesinvolves bulk information, in digital form, and is described byrepresenting each picture element (pixel) in a picture (or image frame)with 8 bits (1 Byte). Aggregation of uncompressed video data results invery large bit quantities, and as a consequence demands large bandwidthallocation for transmission over conventional communication networks inreal time due to limited bandwidth. Due to significant reduncancy inimages between successive frames, data compression is freely applied inreal time video transmission appliations. Data compression may, however,compromise picture quality and so persistent efforts continue to be madeto develop data compression techniques allowing real time transmissionof high quality video over bandwidth limited resources.

In video compression systems, an objective is to represent the videoinformation with as little “capacity” as possible, where capacity isusually measured in bits, either as a constant value or as bits/timeunit. By minimizing bits, the amount of bits that need to be transmittedis reduced, and therefore, the amount of, communication resources neededto support the real time transmission of video data is also reduced.

The most common video coding methods are described in the MPEG* (e.g.,MPEG 2 and MPEG 3) and H.26* (e.g., H.263 and H.264) standards.According to these standards, the video data is exposed to four mainprocesses before transmission, namely prediction, transformation,quantization and entropy coding.

The prediction process performed in a prediction processor significantlyreduces the number of bits required for each frame in a video sequenceto be transferred. It takes advantage of the similarity of parts of thesequence with other parts of the sequence. A decoder that decodes thebit stream has side information to assist in the decoding process. Thisside information is known to both encoder and decoder and so only thedifference has to be transferred. This difference typically requiresmuch less capacity for its representation than the full image. Themotion estimation aspect of the prediction is mainly based on picturecontent from previously reconstructed pictures where the location of thecontent is defined by motion vectors. The prediction process istypically performed on square block sizes (e.g. 16×16 pixels), althoughthe size of the blocks may vary.

Note that in some cases, predictions of pixels based on the adjacentpixels in the same picture rather than pixels of preceding pictures areused. This is referred to as intra prediction, as opposed to interprediction.

The transform and quantization processes will now be discussed in moredetail. The residual represented as a block of data (e.g. 4×4 or 8×8pixels) may still contain internal correlation. A conventional method oftaking advantage of this is to perform a two dimensional blocktransform. The ITU recommendation H.264 uses a 4×4 or 8×8 integer typetransform. This transforms n×n pixels into n×n transform coefficientsand they can usually be represented by fewer bits than the raw pixelrepresentation. Transformation of an n×n array of pixels with internalcorrelation will often result in an n×n block of transform coefficientswith much fewer non-zero values than the original n×n pixel block.

Direct representation of the transform coefficients is still too costlyfor many applications. A quantization process is carried out for afurther reduction of the data representation. Hence the transformcoefficients output from the transform undergo quantization. Thepossible value range of the transform coefficients is divided into valueintervals (or gradations), each limited by an uppermost and lowermostdecision value and assigned a fixed quantization value. The transformcoefficients are then quantified to the quantization value associatedwith the intervals within which the respective coefficients reside.Coefficients being lower than the lowest decision value are quantifiedto zeros. It should be mentioned that this quantization process resultsin the reconstructed video sequence being somewhat different compared tothe uncompressed sequence.

Summarized, a digital video picture is exposed to the following steps:

-   -   Divide the picture into square blocks of pixels, for instance        16×16 or 8×8 pixels. This is done for luminance information as        well as for chrominance information.    -   Produce a prediction for the pixels in the block. This may be        based on pixels in an already coded/decoded picture (called        inter prediction) or on already coded/decoded pixels in the same        picture (intra prediction).    -   Form a difference between the pixels to be coded and the        predicted pixels. This is often referred to as a residual.    -   Perform a two dimensional transformation of the residual        resulting in a representation as transform coefficients.    -   Perform a quantization of the transform coefficients. This is        the major tool for controlling the bit production and        reconstructed picture quality.    -   Establish a scanning of the two dimensional transform        coefficient data into a one dimensional set of data.    -   Perform lossless entropy coding of the quantized transform        coefficients.

The above steps are listed in a natural order for the encoder. Thedecoder will to some extent perform the operations in the opposite orderand do “inverse” operations as inverse transform instead of transformand de-quantization instead of quantization.

SUMMARY

The following paragraphs have been provided by way of generalintroduction, and are not intended to limit the scope of the appendedclaims. The described embodiments, together with further advantages,will be best understood by reference to the following detaileddescription taken in conjunction with the accompanying drawings.

The present inventor recognized limitations with conventional transformdevices and methods in that they do not employ matrices that strike adesirable balance between vector length (and corresponding block size),transform element size, vector norms below a predetermined threshold,and inner products being less than a predetermined threshold.

In addressing these and other limitations, one aspect of the presentdisclosure provides a method (and device and computer program product)of video coding for transforming a first block of residual pixel valuesof a moving picture by a transform matrix formed by a number of basisvectors generating a corresponding second block of transformcoefficients. The transform matrix may be derived from a discrete cosinetransform (DCT) or Karhunen-Loeve transform (KLT). The basis vectors areclose to, but need not be exactly orthogonal. The norms of the basisvectors have a deviation less than 0.0015(or 0.15%), and the elements ofthe basis vectors are representable by 1 byte (elements in an inclusiverange of −128 through 127).

Other aspects include a corresponding inverse method, device andcomputer program product, as well as video equipment, such as a videoteleconferencing endpoint, that hosts the transform.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects and advantages of the presentinvention will become better understood with regard to the followingdescription, appended claims and accompanying drawings where:

FIG. 1 is a block diagram showing a process flow in a video encoder thatincludes a transform operation according to the present disclosure.

FIG. 2 is a block diagram showing a process flow in a video decoder thatincludes an inverse transform operation according to the presentdisclosure.

FIG. 3 is a block diagram representing the transform process flowemployed in the exemplary embodiments; and

FIG. 4 is a block diagram of an exemplary image processing device thatcan be used to perform or implement the method, device and computerprogram product that employs the transformation encoding and/or decodingdescribed herein.

DETAILED DESCRIPTION

In FIG. 1, a current frame as well as one or more reference frames areinput to a prediction processor 1. The prediction processor 1 includesan intra prediction processing path 3 and a motion estimation processingpath 5, the selection of which is controlled by switch 7. Intraprediction processing is selected for fiding similarities within thecurrent image frame, and is thus referred to as “intra” prediction.Motion estimation has a temporal component and thus involves anslyaisbetween successive frames is referred to as “inter” prediction.

The output of the switch 7 is combined with the pixels of the currentframe in a combiner 9, prior to being subjected to core coding 11, whichincludes a transform processor 13 (often involving rate distortioncontrol). Typically the transform is a discrete cosine transform (DCT),which is generally considered to a very good transform to be used inimage and video compression. The DCT is defined with real numbers. Forpractical implementations there are good reasons to use a transform withinteger numbers instead, as will be discussed later. The transformeddata is then subjected to quantization in a quantizer 15 and thensubject to an entropy encoder 17. Entropy encoding removes redundancieswithout losing information, and is referred to as a lossless encodingprocess. Subsequently, the encoded data is arranged in network packetsvia a packetizer 19, prior to be transmitted in a bit stream. Thetransform matices described herein are employed in the transformprocessor 13. Likewise, an inverse quantization and transform isperformed at a decoder to recover the image.

FIG. 2 is a block diagram showing a process flow within a decoder deviceaccording to an embodiment. An input bit stream from an encoder isapplied to a depacketizer and decoder 22, which unpackages the databefore sending the data to an inverse quantizer 24. The inversequantizer 24 provides a best-effort at providing an inverse of thequantization performed in the quantization 15 (FIG. 1). The output ofthe inverse quantization 24 is applied to an inverse integer transform26, which performs an inverse to the integer transform performed in thevideo encoder (FIG. 1). The integer transform 26 uses the same vectorsas in the transform matrix to arrive at the original residue data, whichwas originally applied to the transform maxtrix in the encoder. Theresidue dat is then subjected to a reconstruction operation using motioncompesation input to produce an output video stream.

Most conventional video standards produce block-based residuals thatundergo a two dimensional transform on the encoding side and acorresponding inverse transform on the decoding side, and separabletransforms are usually used. This means that a one dimensional transformis first performed in one direction (e.g. horizontal) followed by thesame one dimensional transform being used on the result in the otherdirection (e.g., vertical). Hence only one dimensional transforms haveto be defined and used.

In the standards H.261, H.263, and the corresponding MPEG standards, a 8point DCT is used. According to signal theory, the DCT is close to anoptimal transform if the correlation between neighbouring pixels in theresidual signal is relatively high. The DCT is defined as 8 real valuedbasis vectors as listed below with 3 digit accuracy:

-   -   0.354 0.354 0.354 0.354 0.354 0.354 0.354 0.354    -   0.490 0.416 0.278 0.098-0.098-0.278-0.416-0.490    -   0.462 0.191-0.191-0.462-0.462-0.191 0.191 0.462    -   0.416-0.098-0.490-0.278 0.278 0.490 0.098-0.416    -   0.354-0.354-0.354 0.354 0.354-0.354-0.354 0.354    -   0.278-0.490 0.098 0.416-0.416-0.098 0.490-0.278    -   0.191-0.462 0.462-0.191-0.191 0.462-0.462 0.191    -   0.098-0.278 0.416-0.490 0.490-0.416 0.278-0.098

These basis vectors are said to be orthonormal, which means that (1)each vector (e.g., row) is orthogonal to all the other vectors, and (2)the norm, or square sum along each line, is equal to 1.

Defining now the above matrix of data T(i,j), i,j=0-7. i indicates thecolumn, number and j indicates row number. Assuming further that thereis a set of 8 pixel values f(i) of an 8×8 block to be transformed into 8transform values F(j). The following relations then apply:

${{F(j)} = {\sum\limits_{{i = 0},7}{{f(i)}*{T\left( {i,j} \right)}}}},{j = {0 - 7}}$Inverse transform or reconstruction of the pixels:

${{f(i)} = {\sum\limits_{{j = 0},7}{{F(j)}*{T\left( {i,j} \right)}}}},{i = {0 - 7}}$The transform is then normally preformed on each set of 8 pixel valuesof the 8×8 block, and then the whole procedure is repeated on the resultvertically with the same transform matrix.

This transform performs well in theory. However in practice, asrecognized by the present inventor, it has a disadvantage that it isbased on real number representation. In implementation this willinevitably result in some inaccuracy, which in turn will be slightlydifferent for different implementations using different internalrepresentations of real numbers. This may lead to a mismatch betweendifferent reconstruction procedures. The implementation is alsorelatively complex. For these reasons it is desirable to use transformsthat are described by integer numbers only.

The orthonormal conditions defined above are still desirable. Withinteger numbers it is impossible to obtain the condition that the squaresum along each line is equal to 1. It is therefore more realistic andfeasible to change the requirement of the norm to be the same for allvectors, but still maintaining the first requirement that each vectorshall be orthogonal to all the other vectors.

An example of basis vectors satisfying the last-mentioned requirementsare the Walsh Hadamard vectors. 8 point Hadamard vectors are as follows:

-   1 1 1 1 1 1 1 1-   1 1 1 1-1-1-1-1-   1 1-1-1-1-1 1 1-   1 1-1-1 1 1-1-1-   1-1-1 1 1-1-1 1-   1-1-1 1-1 1 1-1-   1-1 1-1-1 1-1 1-   1-1 1-1 1-1 1-1

However, this transform turns out to have other drawbacks and istherefore not very much used.

A better and expanded set of requirements for integer transforms isfound via the application of the following criteria:

-   -   1. The vectors shall be orthogonal.    -   2. The numbers used to define the vectors are small—typically        less than 32, although as will be discussed, but also within a        range representable in 1 Byte.    -   3. The shape of the vectors are as close as possible to the        corresponding DCT.    -   4. The norm, or square sum along each line shall be the same for        all vectors.

It is difficult to satisfy all the requirements—especially for largertransforms. The above requirements can therefore be considered to belisted in prioritized order. Traditionally, 1 and 2 have been consideredto be mandatory, 3 may be only partly fulfilled whereas 4 may be workedaround with some added complexity in the implementations.

In the early development of H.264 the following transform wasconsidered, but not adopted:

-   13 13 13 13-   17 7-7-17-   13-13-13 13-   7-17 17-7

This vector set satisfied requirements 1,2,3,4 above. H.264 and MPEG AVCstandards use both 4 point and 8 point integer transforms. As for the 4point transform, the basis vectors are:

-   1 1 1 1-   2 1-1-2-   1-1-1 1-   1-2 2-1    Although this matrix was adopted in H.264, the vectors do not have    the same norm, or close to the same norm. As a consequence, more    numeric manipulation is required during quantization to avoid bias.

This vector set fulfils requirements 1, 2 and partly 3 above. As for the8 point transform, the basis vectors are:

-   1 1 1 1 1 1 1 1-   12 10 6 3-3-6-10-12-   2 1-1-2-2-1 1 2-   10-3-12-6 6 12 3-10-   1-1-1 1 1-1-1 1-   6-12 3 10-10-3 12-6-   1-2 2-1-1 2-2 1-   3-6 10-12 12-10 6-3

This vector set also satisfies requirements 1, 2 and partly 3. However,three different norms are present, with fairly large differences invalue. Inventor experience has shown that it is still possible to reducethe requirements even further to improve coding efficiency and reducethe complexity without compromising with picture quality. Therequirements for the transform are modified from above:

-   -   1. The vectors may not be exactly orthogonal, but very close.    -   2. The numbers used to define the vectors are small—say <32, or        even 127, so as to be represented in one Byte.    -   3. The shape of the vectors are as close as possible to the        optimal transform. Depending on the statistics of the signal to        be coded, this may be DCT or a Karhunen-Loeve (KLT) transform        based on the statistics of the signal.    -   4. The norm of the vectors may not be exactly the same, but very        close.

The present inventor realized that the relaxations on 1 and 4 may besmall enough so that the encoding/decoding performance is practicallythe same as if 1 and 4 were completely fulfilled.

The present embodiment employs a design of transform vector sets withslightly relaxed requirements 1 and 4 resulting in better codingperformance mainly due to better fulfilment of requirement 3. The basisvectors have a shape close to a corresponding DCT or KLT, and at thesame time norms that are similar but not necessarily exactly equal witha maximum deviation less than 0.01, or better 0.0015 relative to the DCvector (the vector with all elements have a common sign).

As a measure for the deviation from the DCT, it should be understoodthat for an N-point DCT the basis vectors V(i,j) may be written as:

Transform coefficient j:V(i,j)=Kj*cos(π/2N*(j−1)*(2i−1)), i,j=1 to N.Kj is a normalization constant for the transform coefficient j. Thecomparison is to be made between the vector sets U(i,j) and V(i,j). Itis assumed that Kj is chosen so that the norm of the vector set V(i,j)is the same as for the vector set U(i,j).

A measure for the difference between vector elements in the twotransforms is an “element deviation”:D(i,j)=|V(i,j)−U(i,j)|/sqrt(Norm(j)/N).It is then possible to specify a set of vectors that have a shape thatis close to the shape of the DCT vectors if all values D(i,j) for a setof vectors have an element deviation smaller than or equal to 0.05, forexample.

In addition, the basis vectors are close to orthogonal but not exactlyorthogonal, and the numbers used to specify the set of transform vectorsare small, and in a range of −128 to 127, so as to be represented in onebyte.

A first step for approaching a set of transform vectors according to thepresent invention is to make an integer version of the DCT vectors bymultiplying the DCT vectors by 26*√2, a scaling factor, and rounding tothe nearest integer. The result would be as follows, with the square sumto the right of each row:

13 13 13 13 13 13 13 13 1352 18 15 10 4 −4 −10 −15 −18 1330 17 7 −7 −17−17 −7 7 17 1352 15 −4 −18 −10 10 18 4 −15 1330 13 −13 −13 13 13 −13 −1313 1352 10 −18 4 15 −15 −4 18 −10 1330 7 −17 17 −7 −7 17 −17 7 1352 4−10 15 −18 18 −15 10 −4 1330

The square sum indicates that the norm is not identical for all vectors,nor is it within the 0.0015 threshold. However, as a second step forapproaching a set of transform vectors according to the presentembodiment, a slight modification of the numbers can make the norm ofeach vector much closer to each other, and close enough to be consideredto have equal norm. An approximate orthogonality between vectors, closeenough to be considered as orthogonal can also be obtained, in additionto a shape of the vectors very close to the DCT can also be obtained bya slight modification of the numbers.

The inventor has obtained the following set of vectors as an example ofan appropriate transform according to the present embodiment (the numberon the right side of each row is the square sum of the respectivevector):

13 13 13 13 13 13 13 13 1352 17 16 11 3 −3 −11 −16 −17 1350 17 7 −7 −17−17 −7 7 17 1352 16 −3 −17 −11 11 17 3 −16 1350 13 −13 −13 13 13 −13 −1313 1352 11 −17 3 16 −16 −3 17 −11 1350 7 −17 17 −7 −7 17 −17 7 1352 3−11 16 −17 17 −16 11 −3 1350

In the decoding process, to obtain residual pixels from transformcoefficients, the inverse set of vectors are used, i.e. the transpose ofthe set of vectors above is used. The decoder may be implemented in aprogrammed processor, firmware circuit or dedicated circuit device, forexample.

As further refinement, alternative transform matrixes are presented thatcomply with the criteria described below. Because the transform usesinteger elements, the transform cannot be identical to the DCT. For thedescription below, it will be referred to as DCT', indicating that it isan approximation to the real DCT due to the integer constraint. AnN-point DCT′ transformation may be defined by a set of N basis vectorsU(i,j)={x1j, x2j, x3j, . . . xNj}, j=1. . . N. In one and only one ofthe vectors, all the elements xi,j have the same sign. This vector iscalled the DC vector. The norm of a vector is defined asNorm(j)=sum_i=1,N(xij*xij). Then the inner product of two vectors isdefined as Inner(j0,j1)=sum_i=1,N(xij0*xij1). If Inner(j0,j1)=0 thevectors j0 and j1 are defined to be orthogonal. Consistent with thisnomenclature, a transform consistent with the principles of the presentembodiment will comply with the following requirements:

-   -   1. All elements xij shall be integer numbers.    -   2. All elements xij shall be in the range <−128 to 127>, so as        to fit into 1 Byte.    -   3. None of the vectors shall have a norm that differs more than        0.0015 from the norm of the DC vector.    -   4. For all j0,j1 where j0 is different from j1, |Inner(j0,j1)|        shall be less that 0.003.    -   5. The vectors shall have a shape that is close to the shape of        the DCT vectors, such as a maximum element deviation for any        vector element is not greater than 0.05.

The following is an example of a 4×4 matrix that complies with the aboverequirements. Each vector is contained within a pair of brackets. Thetransform matrix may be applied in an encoding context or decodingcontext, recognizing that the vector basis sets are the same and so thefollow matrix may also be recognized as an inverse transform matrix usedin a decoding setting.

-   {64, 64, 64, 64,}-   {83, 36, -36, -83,}-   {64, -64, -64, 64,}-   {36, -83, 83, -36,}

The following is an example of a 8×8 matrix that complies with the aboverequirements. Each vector is contained within a pair of brackets. Thetransform matrix may be applied in an encoding context or decodingcontext, recognizing that the vector basis sets are the same and so thefollow matrix may also be recognized as an inverse transform matrix usedin a decoding setting.

-   {64, 64, 64, 64, 64, 64, 64, 64,}-   {89, 75, 50, 18, -18, -50, -75, -89,}-   {83, 36, -36, -83, -83, -36, 36, 83,}-   {75, -18, -89, -50, 50, 89, 18, -75,}-   {64, -64, -64, 64, 64, -64, -64, 64,}-   {50, -89, 18, 75, -75, -18, 89, -50,}-   {36, -83, 83, -36, -36, 83, -83, 36,}-   {18, -50, 75, -89, 89, -75, 50, -18,}

The following is an example of a 16×16 matrix that complies with theabove requirements. Each vector is contained within a pair of brackets.The transform matrix may be applied in an encoding context or decodingcontext, recognizing that the vector basis sets are the same and so thefollow matrix may also be recognized as an inverse transform matrix usedin a decoding setting.

-   {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,}-   {90, 87, 80, 70, 57, 43, 25, 9, -9, -25, -43, -57, -70, -80, -87,    -90,}-   {89, 75, 50, 18, -18, -50, -75, -89, -89, -75, -50, -18, 18, 50, 75,    89,}-   {87, 57, 9, -43, -80, -90, -70, -25, 25, 70, 90, 80, 43, -9, -57,    -87,}-   {83, 36, -36, -83, -83, -36, 36, 83, 83, 36, -36, -83, -83, -36, 36,    83,}-   {80, 9, -70, -87, -25, 57, 90, 43, -43, -90, -57, 25, 87, 70, -9,    -80,}-   {75, -18, -89, -50, 50, 89, 18, -75, -75, 18, 89, 50, -50, -89, -18,    75,}-   {70, -43, -87, 9, 90, 25, -80, -57, 57, 80, -25, -90, -9, 87, 43,    -70,}-   {64, -64, -64, 64, 64, -64, -64, 64, 64, -64, -64, 64, 64, -64, -64,    64,}-   {57, -80, -25, 90, -9, -87, 43, 70, -70, -43, 87, 9, -90, 25, 80,    -57,}-   {50, -89, 18, 75, -75, -18, 89, -50, -50, 89, -18, -75, 75, 18, -89,    50,}-   {43, -90, 57, 25, -87, 70, 9, -80, 80, -9, -70, 87, -25, -57, 90,    -43,}-   {36, -83, 83, -36, -36, 83, -83, 36, 36, -83, 83, -36, -36, 83, -83,    36,}-   {25, -70, 90, -80, 43, 9, -57, 87, -87, 57, -9, -43, 80, -90, 70,    -25,}-   {18, -50, 75, -89, 89, -75, 50, -18, -18, 50, -75, 89, -89, 75, -50,    18,}-   {9, -25, 43, -57, 70, -80, 87, -90, 90, -87, 80, -70, 57, -43, 25,    -9,}

The following is an example of a 32×32 matrix that complies with theabove requirements. Each vector is contained within a pair of brackets.The transform matrix may be applied in an encoding context or decodingcontext, recognizing that the vector basis sets are the same and so thefollow matrix may also be recognized as an inverse transform matrix usedin a decoding setting.

-   {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,    64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,}-   {90, 90, 88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4, -4,    -13, -22, -31, -38, -46, -54, -61, -67, -73, -78, -82, -85, -88,    -90, -90,}-   {90, 87, 80, 70, 57, 43, 25, 9, -9, -25, -43, -57, -70, -80, -87,    -90, -90, -87, -80, -70, -57, -43, -25, -9, 9, 25, 43, 57, 70, 80,    87, 90,}-   {90, 82, 67, 46, 22, -4, -31, -54, -73, -85, -90, -88, -78, -61,    -38, -13, 13, 38, 61, 78, 88, 90, 85, 73, 54, 31, 4, -22, -46, -67,    -82, -90,}-   {89, 75, 50, 18, -18, -50, -75, -89, -89, -75, -50, -18, 18, 50, 75,    89, 89, 75, 50, 18, -18, -50, -75, -89, -89, -75, -50, -18, 18, 50,    75, 89,}-   {88, 67, 31, -13, -54, -82, -90, -78, -46, -4, 38, 73, 90, 85, 61,    22, -22, -61, -85, -90, -73, -38, 4, 46, 78, 90, 82, 54, 13, -31,    -67, -88,}-   {87, 57, 9, -43, -80, -90, -70, -25, 25, 70, 90, 80, 43, -9, -57,    -87, -87, -57, -9, 43, 80, 90, 70, 25, -25, -70, -90, -80, -43, 9,    57, 87,}-   {85, 46, -13, -67, -90, -73, -22, 38, 82, 88, 54, -4, -61, -90, -78,    -31, 31, 78, 90, 61, 4, -54, -88, -82, -38, 22, 73, 90, 67, 13, -46,    -85,}-   {83, 36, -36, -83, -83, -36, 36, 83, 83, 36, -36, -83, -83, -36, 36,    83, 83, 36, -36, -83, -83, -36, 36, 83, 83, 36, -36, -83, -83, -36,    36, 83,}-   {82, 22, -54, -90, -61, 13, 78, 85, 31, -46, -90, -67, 4, 73, 88,    38, -38, -88, -73, -4, 67, 90, 46, -31, -85, -78, -13, 61, 90, 54,    -22, -82,}-   {80, 9, -70, -87, -25, 57, 90, 43, -43, -90, -57, 25, 87, 70, -9,    -80, -80, -9, 70, 87, 25, -57, -90, -43, 43, 90, 57, -25, -87, -70,    9, 80,}-   {78, -4, -82, -73, 13, 85, 67, -22, -88, -61, 31, 90, 54, -38, -90,    -46, 46, 90, 38, -54, -90, -31, 61, 88, 22, -67, -85, -13, 73, 82,    4, -78,}-   {75, -18, -89, -50, 50, 89, 18, -75, -75, 18, 89, 50, -50, -89, -18,    75, 75, -18, -89, -50, 50, 89, 18, -75, -75, 18, 89, 50, -50, -89,    -18, 75,}-   {73, -31, -90, -22, 78, 67, -38, -90, -13, 82, 61, -46, -88, -4, 85,    54, -54, -85, 4, 88, 46, -61, -82, 13, 90, 38, -67, -78, 22, 90, 31,    -73 ,}-   {70, -43, -87, 9, 90, 25, -80, -57, 57, 80, -25, -90, -9, 87, 43,    -70, -70, 43, 87, -9, -90, -25, 80, 57, -57, -80, 25, 90, 9, -87,    -43, 70,}-   {67, -54, -78, 38, 85, -22, -90, 4, 90, 13, -88, -31, 82, 46, -73,    -61, 61, 73, -46, -82, 31, 88, -13, -90, -4, 90, 22, -85, -38, 78,    54, -67,}-   {64, -64, -64, 64, 64, -64, -64, 64, 64, -64, -64, 64, 64, -64, -64,    64, 64, -64, -64, 64, 64, -64, -64, 64, 64, -64, -64, 64, 64, -64,    -64, 64,}-   {61, -73, -46, 82, 31, -88, -13, 90, -4, -90, 22, 85, -38, -78, 54,    67, -67, -54, 78, 38, -85, -22, 90, 4, -90, 13, 88, -31, -82, 46,    73, -61,}-   {57, -80, -25, 90, -9, -87, 43, 70, -70, -43, 87, 9, -90, 25, 80,    -57, -57, 80, 25, -90, 9, 87, -43, -70, 70, 43, -87, -9, 90, -25,    -80, 57 ,}-   {54, -85, -4, 88, -46, -61, 82, 13, -90, 38, 67, -78, -22, 90, -31,    -73, 73, 31, -90, 22, 78, -67, -38, 90, -13, -82, 61, 46, -88, 4,    85, -54,}-   {50, -89, 18, 75, -75, -18, 89, -50, -50, 89, -18, -75, 75, 18, -89,    50, 50, -89, 18, 75, -75, -18, 89, -50, -50, 89, -18, -75, 75, 18,    -89, 50,}-   {46, -90, 38, 54, -90, 31, 61, -88, 22, 67, -85, 13, 73, -82, 4, 78,    -78, -4, 82, -73, -13, 85, -67, -22, 88, -61, -31, 90, -54, -38, 90,    -46,}-   {43, -90, 57, 25, -87, 70, 9, -80, 80, -9, -70, 87, -25, -57, 90,    -43, -43, 90, -57, -25, 87, -70, -9, 80, -80, 9, 70, -87, 25, 57,    -90, 43,}-   {38, -88, 73, -4, -67, 90, -46, -31, 85, -78, 13, 61, -90, 54, 22,    -82, 82, -22, -54, 90, -61, -13, 78, -85, 31, 46, -90, 67, 4, -73,    88, -38,}-   {36, -83, 83, -36, -36, 83, -83, 36, 36, -83, 83, -36, -36, 83, -83,    36, 36, -83, 83, -36, -36, 83, -83, 36, 36, -83, 83, -36, -36, 83,    -83, 36,}-   {31, -78, 90, -61, 4, 54, -88, 82, -38, -22, 73, -90, 67, -13, -46,    85, -85, 46, 13, -67, 90, -73, 22, 38, -82, 88, -54, -4, 61, -90,    78, -31,}-   {25, -70, 90, -80, 43, 9, -57, 87, -87, 57, -9, -43, 80, -90, 70,    -25, -25, 70, -90, 80, -43, -9, 57, -87, 87, -57, 9, 43, -80, 90,    -70, 25,}-   {22, -61, 85, -90, 73, -38, -4, 46, -78, 90, -82, 54, -13, -31, 67,    -88, 88, -67, 31, 13, -54, 82, -90, 78, -46, 4, 38, -73, 90, -85,    61, -22,}-   {18, -50, 75, -89, 89, -75, 50, -18, -18, 50, -75, 89, -89, 75, -50,    18, 18, -50, 75, -89, 89, -75, 50, -18, -18, 50, -75, 89, -89, 75,    -50, 18,}-   {13, -38, 61, -78, 88, -90, 85, -73, 54, -31, 4, 22, -46, 67, -82,    90, -90, 82, -67, 46, -22, -4, 31, -54, 73, -85, 90, -88, 78, -61,    38, -13,}-   {9, -25, 43, -57, 70, -80, 87, -90, 90, -87, 80, -70, 57, -43, 25,    -9, -9, 25, -43, 57, -70, 80, -87, 90, -90, 87, -80, 70, -57, 43,    -25, 9,}-   {4, -13, 22, -31, 38, -46, 54, -61, 67, -73, 78, -82, 85, -88, 90,    -90, 90, -90, 88, -85, 82, -78, 73, -67, 61, -54, 46, -38, 31, -22,    13, -4,}

The following is an example of a 64×64 matrix that complies with theabove requirements. Each vector is contained within a pair of brackets.The transform matrix may be applied in an encoding context or decodingcontext, recognizing that the vector basis sets are the same and so thefollow matrix may also be recognized as an inverse transform matrix usedin a decoding setting.

-   {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,    64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,    64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,    64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,}-   {91, 90, 90, 89, 88, 87, 86, 84, 83, 81, 79, 76, 74, 71, 69, 66, 62,    59, 56, 52, 48, 45, 41, 37, 33, 28, 24, 20, 16, 11, 7, 2, -2, -7,    -11, -16, -20, -24, -28, -33, -37, -41, -45, -48, -52, -56, -59,    -62, -66, -69, -71, -74, -76, -79, -81, -83, -84, -86, -87, -88,    -89, -90, -90, -91,}-   {90, 90, 88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4, -4,    -13, -22, -31, -38, -46, -54, -61, -67, -73, -78, -82, -85, -88,    -90, -90, -90, -90, -88, -85, -82, -78, -73, -67, -61, -54, -46,    -38, -31, -22, -13, -4, 4, 13, 22, 31, 38, 46, 54, 61, 67, 73, 78,    82, 85, 88, 90, 90,}-   {90, 88, 84, 79, 71, 62, 52, 41, 28, 16, 2, -11, -24, -37, -48, -59,    -69, -76, -83, -87, -90, -91, -89, -86, -81, -74, -66, -56, -45,    -33, -20, -7, 7, 20, 33, 45, 56, 66, 74, 81, 86, 89, 91, 90, 87, 83,    76, 69, 59, 48, 37, 24, 11, -2, -16, -28, -41, -52, -62, -71, -79,    -84, -88, -90,}-   {90, 87, 80, 70, 57, 43, 25, 9, -9, -25, -43, -57, -70, -80, -87,    -90, -90, -87, -80, -70, -57, -43, -25, -9, 9, 25, 43, 57, 70, 80,    87, 90, 90, 87, 80, 70, 57, 43, 25, 9, -9, -25, -43, -57, -70, -80,    -87, -90, -90, -87, -80, -70, -57, -43, -25, -9, 9, 25, 43, 57, 70,    80, 87, 90,}-   {90, 84, 74, 59, 41, 20, -2, -24, -45, -62, -76, -86, -90, -89, -83,    -71, -56, -37, -16, 7, 28, 48, 66, 79, 87, 91, 88, 81, 69, 52, 33,    11, -11, -33, -52, -69, -81, -88, -91, -87, -79, -66, -48, -28, -7,    16, 37, 56, 71, 83, 89, 90, 86, 76, 62, 45, 24, 2, -20, -41, -59,    -74, -84, -90,}-   {90, 82, 67, 46, 22, -4, -31, -54, -73, -85, -90, -88, -78, -61,    -38, -13, 13, 38, 61, 78, 88, 90, 85, 73, 54, 31, 4, -22, -46, -67,    -82, -90, -90, -82, -67, -46, -22, 4, 31, 54, 73, 85, 90, 88, 78,    61, 38, 13, -13, -38, -61, -78, -88, -90, -85, -73, -54, -31, -4,    22, 46, 67, 82, 90,}-   {89, 79, 59, 33, 2, -28, -56, -76, -88, -90, -81, -62, -37, -7, 24,    52, 74, 87, 90, 83, 66, 41, 11, -20, -48, -71, -86, -91, -84, -69,    -45, -16, 16, 45, 69, 84, 91, 86, 71, 48, 20, -11, -41, -66, -83,    -90, -87, -74, -52, -24, 7, 37, 62, 81, 90, 88, 76, 56, 28, -2, -33,    -59, -79, -89,}-   {89, 75, 50, 18, -18, -50, -75, -89, -89, -75, -50, -18, 18, 50, 75,    89, 89, 75, 50, 18, -18, -50, -75, -89, -89, -75, -50, -18, 18, 50,    75, 89, 89, 75, 50, 18, -18, -50, -75, -89, -89, -75, -50, -18, 18,    50, 75, 89, 89, 75, 50, 18, -18, -50, -75, -89, -89, -75, -50, -18,    18, 50, 75, 89,}-   {88, 71, 41, 2, -37, -69, -87, -89, -74, -45, -7, 33, 66, 86, 90,    76, 48, 11, -28, -62, -84, -90, -79, -52, -16, 24, 59, 83, 91, 81,    56, 20, -20, -56, -81, -91, -83, -59, -24, 16, 52, 79, 90, 84, 62,    28, -11, -48, -76, -90, -86, -66, -33, 7, 45, 74, 89, 87, 69, 37,    -2, -41, -71, -88,}-   {88, 67, 31, -13, -54, -82, -90, -78, -46, -4, 38, 73, 90, 85, 61,    22, -22, -61, -85, -90, -73, -38, 4, 46, 78, 90, 82, 54, 13, -31,    -67, -88, -88, -67, -31, 13, 54, 82, 90, 78, 46, 4, -38, -73, -90,    -85, -61, -22, 22, 61, 85, 90, 73, 38, -4, -46, -78, -90, -82, -54,    -13, 31, 67, 88,}-   {87, 62, 20, -28, -69, -89, -84, -56, -11, 37, 74, 90, 81, 48, 2,    -45, -79, -91, -76, -41, 7, 52, 83, 90, 71, 33, -16, -59, -86, -88,    -66, -24, 24, 66, 88, 86, 59, 16, -33, -71, -90, -83, -52, -7, 41,    76, 91, 79, 45, -2, -48, -81, -90, -74, -37, 11, 56, 84, 89, 69, 28,    -20, -62, -87,}-   {87, 57, 9, -43, -80, -90, -70, -25, 25, 70, 90, 80, 43, -9, -57,    -87, -87, -57, -9, 43, 80, 90, 70, 25, -25, -70, -90, -80, -43, 9,    57, 87, 87, 57, 9, -43, -80, -90, -70, -25, 25, 70, 90, 80, 43, -9,    -57, -87, -87, -57, -9, 43, 80, 90, 70, 25, -25, -70, -90, -80, -43,    9, 57, 87,}-   {86, 52, -2, -56, -87, -84, -48, 7, 59, 88, 83, 45, -11, -62, -89,    -81, -41, 16, 66, 90, 79, 37, -20, -69, -90, -76, -33, 24, 71, 91,    74, 28, -28, -74, -91, -71, -24, 33, 76, 90, 69, 20, -37, -79, -90,    -66, -16, 41, 81, 89, 62, 11, -45, -83, -88, -59, -7, 48, 84, 87,    56, 2, -52, -86,}-   {85, 46, -13, -67, -90, -73, -22, 38, 82, 88, 54, -4, -61, -90, -78,    -31, 31, 78, 90, 61, 4, -54, -88, -82, -38, 22, 73, 90, 67, 13, -46,    -85, -85, -46, 13, 67, 90, 73, 22, -38, -82, -88, -54, 4, 61, 90,    78, 31, -31, -78, -90, -61, -4, 54, 88, 82, 38, -22, -73, -90, -67,    -13, 46, 85,}-   {84, 41, -24, -76, -89, -56, 7, 66, 91, 69, 11, -52, -88, -79, -28,    37, 83, 86, 45, -20, -74, -90, -59, 2, 62, 90, 71, 16, -48, -87,    -81, -33, 33, 81, 87, 48, -16, -71, -90, -62, -2, 59, 90, 74, 20,    -45, -86, -83, -37, 28, 79, 88, 52, -11, -69, -91, -66, -7, 56, 89,    76, 24, -41, -84,}-   {83, 36, -36, -83, -83, -36, 36, 83, 83, 36, -36, -83, -83, -36, 36,    83, 83, 36, -36, -83, -83, -36, 36, 83, 83, 36, -36, -83, -83, -36,    36, 83, 83, 36, -36, -83, -83, -36, 36, 83, 83, 36, -36, -83, -83,    -36, 36, 83, 83, 36, -36, -83, -83, -36, 36, 83, 83, 36, -36, -83,    -83, -36, 36, 83,}-   {83, 28, -45, -88, -74, -11, 59, 91, 62, -7, -71, -89, -48, 24, 81,    84, 33, -41, -87, -76, -16, 56, 90, 66, -2, -69, -90, -52, 20, 79,    86, 37, -37, -86, -79, -20, 52, 90, 69, 2, -66, -90, -56, 16, 76,    87, 41, -33, -84, -81, -24, 48, 89, 71, 7, -62, -91, -59, 11, 74,    88, 45, -28, -83,}-   {82, 22, -54, -90, -61, 13, 78, 85, 31, -46, -90, -67, 4, 73, 88,    38, -38, -88, -73, -4, 67, 90, 46, -31, -85, -78, -13, 61, 90, 54,    -22, -82, -82, -22, 54, 90, 61, -13, -78, -85, -31, 46, 90, 67, -4,    -73, -88, -38, 38, 88, 73, 4, -67, -90, -46, 31, 85, 78, 13, -61,    -90, -54, 22, 82,}-   {81, 16, -62, -90, -45, 37, 88, 69, -7, -76, -84, -24, 56, 91, 52,    -28, -86, -74, -2, 71, 87, 33, -48, -90, -59, 20, 83, 79, 11, -66,    -89, -41, 41, 89, 66, -11, -79, -83, -20, 59, 90, 48, -33, -87, -71,    2, 74, 86, 28, -52, -91, -56, 24, 84, 76, 7, -69, -88, -37, 45, 90,    62, -16, -81,}-   {80, 9, -70, -87, -25, 57, 90, 43, -43, -90, -57, 25, 87, 70, -9,    -80, -80, -9, 70, 87, 25, -57, -90, -43, 43, 90, 57, -25, -87, -70,    9, 80, 80, 9, -70, -87, -25, 57, 90, 43, -43, -90, -57, 25, 87, 70,    -9, -80, -80, -9, 70, 87, 25, -57, -90, -43, 43, 90, 57, -25, -87,    -70, 9, 80,}-   {79, 2, -76, -81, -7, 74, 83, 11, -71, -84, -16, 69, 86, 20, -66,    -87, -24, 62, 88, 28, -59, -89, -33, 56, 90, 37, -52, -90, -41, 48,    91, 45, -45, -91, -48, 41, 90, 52, -37, -90, -56, 33, 89, 59, -28,    -88, -62, 24, 87, 66, -20, -86, -69, 16, 84, 71, -11, -83, -74, 7,    81, 76, -2, -79,}-   {78, -4, -82, -73, 13, 85, 67, -22, -88, -61, 31, 90, 54, -38, -90,    -46, 46, 90, 38, -54, -90, -31, 61, 88, 22, -67, -85, -13, 73, 82,    4, -78, -78, 4, 82, 73, -13, -85, -67, 22, 88, 61, -31, -90, -54,    38, 90, 46, -46, -90, -38, 54, 90, 31, -61, -88, -22, 67, 85, 13,    -73, -82, -4, 78,}-   {76, -11, -86, -62, 33, 90, 45, -52, -89, -24, 69, 83, 2, -81, -71,    20, 88, 56, -41, -91, -37, 59, 87, 16, -74, -79, 7, 84, 66, -28,    -90, -48, 48, 90, 28, -66, -84, -7, 79, 74, -16, -87, -59, 37, 91,    41, -56, -88, -20, 71, 81, -2, -83, -69, 24, 89, 52, -45, -90, -33,    62, 86, 11, -76,}-   {75, -18, -89, -50, 50, 89, 18, -75, -75, 18, 89, 50, -50, -89, -18,    75, 75, -18, -89, -50, 50, 89, 18, -75, -75, 18, 89, 50, -50, -89,    -18, 75, 75, -18, -89, -50, 50, 89, 18, -75, -75, 18, 89, 50, -50,    -89, -18, 75, 75, -18, -89, -50, 50, 89, 18, -75, -75, 18, 89, 50,    -50, -89, -18, 75,}-   {74, -24, -90, -37, 66, 81, -11, -88, -48, 56, 86, 2, -84, -59, 45,    89, 16, -79, -69, 33, 91, 28, -71, -76, 20, 90, 41, -62, -83, 7, 87,    52, -52, -87, -7, 83, 62, -41, -90, -20, 76, 71, -28, -91, -33, 69,    79, -16, -89, -45, 59, 84, -2, -86, -56, 48, 88, 11, -81, -66, 37,    90, 24, -74,}-   {73, -31, -90, -22, 78, 67, -38, -90, -13, 82, 61, -46, -88, -4, 85,    54, -54, -85, 4, 88, 46, -61, -82, 13, 90, 38, -67, -78, 22, 90, 31,    -73, -73, 31, 90, 22, -78, -67, 38, 90, 13, -82, -61, 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FIG. 3 illustrates a process flow for performing the video coding ordecoding in a schematic representation. A first block of residual pixelvalues can be subjected to a transformation by a transform matrix formedby a number of basis vectors, to generate a corresponding second blockof residual pixel values of a moving picture. For the transformation, atransform matrix is used, as described above. The invention isapplicable to any transform size. It is also not limited to closeapproximation to DCT. The approximation may be to other optimal basisvectors like the KLT.

FIG. 4 illustrates a computer system 1201 upon which the method of videocoding or decoding of the present invention may be implemented. Forexample, the computer system 1201 may be used to implement a videoconferencing endpoint that employs a video encoder and decoder thatimplement the teachings of the present disclosure. The computer system1201 includes a bus 1202 or other communication mechanism forcommunicating information, and a processor 1203 coupled with the bus1202 for processing the information. The computer system 1201 alsoincludes a main memory 1204, such as a random access memory (RAM) orother dynamic storage device (e.g., dynamic RAM (DRAM), static RAM(SRAM), and synchronous DRAM (SDRAM)), coupled to the bus 1202 forstoring information and instructions to be executed by processor 1203.In addition, the main memory 1204 may be used for storing temporaryvariables or other intermediate information during the execution ofinstructions by the processor 1203 for the coding or encoding. Thecomputer system 1201 further includes a read only memory (ROM) 1205 orother static storage device (e.g., programmable ROM (PROM), erasablePROM (EPROM), and electrically erasable PROM (EEPROM)) coupled to thebus 1202 for storing static information and instructions for theprocessor 1203. The transformation can be performed on the processor1203 by using the main memory 1204 for data storage during processing.

The computer system 1201 also includes a disk controller 1206 coupled tothe bus 1202 to control one or more storage devices for storinginformation and instructions, such as a magnetic hard disk 1207, and aremovable media drive 1208 (e.g., floppy disk drive, read-only compactdisc drive, read/write compact disc drive, compact disc jukebox, tapedrive, and removable magneto-optical drive). The storage devices may beadded to the computer system 1201 using an appropriate device interface(e.g., small computer system interface (SCSI), integrated deviceelectronics (IDE), enhanced-IDE (E-IDE), direct memory access (DMA), orultra-DMA). The storage devices can be used to store results of thecoding/decoding or data that has to be transformed, and to store thebasic vectors of the transform matrix.

The computer system 1201 may also include special purpose logic devices(e.g., application specific integrated circuits (ASICs)) or configurablelogic devices (e.g., simple programmable logic devices (SPLDs), complexprogrammable logic devices (CPLDs), and field programmable gate arrays(FPGAs)). With such special purpose logic devices, the transformationwith the transform matrix can be hardware-coded to speed up the encodingor coding of the moving pictures, and to reduce the processing requiredon processor 1203.

The computer system 1201 may also include a display controller 1209coupled to the bus 1202 to control a display 1210, such as a cathode raytube (CRT), for displaying information to a computer user. Results ofthe coding or decoding of a moving picture can be displayed on display1210. The computer system includes input devices, such as a keyboard1211 and a pointing device 1212, for interacting with a computer userand providing information to the processor 1203. The pointing device1212, for example, may be a mouse, a trackball, or a pointing stick forcommunicating direction information and command selections to theprocessor 1203 and for controlling cursor movement on the display 1210.In addition, a printer may provide printed listings of data storedand/or generated by the computer system 1201.

The computer system 1201 performs a portion or all of the processingsteps of the transformation in response to the processor 1203 executingone or more sequences of one or more instructions contained in a memory,such as the main memory 1204. Such instructions may be read into themain memory 1204 from another computer readable medium, such as a harddisk 1207 or a removable media drive 1208. One or more processors in amulti-processing arrangement may also be employed to execute thesequences of instructions contained in main memory 1204. In alternativeembodiments, hard-wired circuitry may be used in place of or incombination with software instructions, as described above with respectto the special purpose logic devices. Thus, embodiments are not limitedto any specific combination of hardware circuitry and software.

As stated above, the computer system 1201 includes at least one computerreadable medium or memory for holding instructions programmed accordingto the teachings of the invention and for containing data structures,tables, records, or other data described herein. For example the basicvectors and moving pictures that have to be coded or decoded can bestored on such computer readable media. Examples of computer readablemedia are compact discs, hard disks, floppy disks, tape, magneto-opticaldisks, PROMs (EPROM, EEPROM, flash EPROM), DRAM, SRAM, SDRAM, or anyother magnetic medium, compact discs (e.g., CD-ROM), or any otheroptical medium, punch cards, paper tape, or other physical medium withpatterns of holes, a carrier wave (described below), or any other mediumfrom which a computer can read.

Stored on any one or on a combination of computer readable media, thepresent invention includes software for controlling the computer system1201, for driving a device or devices for implementing the method ofvideo coding or decoding, and for enabling the computer system 1201 tointeract with a human user (e.g., print production personnel). Suchsoftware may include, but is not limited to, device drivers, operatingsystems, development tools, and applications software. Such computerreadable media further includes the computer program product of thepresent invention for performing all or a portion (if processing isdistributed) of the processing performed in implementing the invention.

The computer code devices of the present invention may be anyinterpretable or executable code mechanism, including but not limited toscripts, interpretable programs, dynamic link libraries (DLLs), Javaclasses, and complete executable programs. Moreover, parts of theprocessing of the present invention may be distributed for betterperformance, reliability, and/or cost.

The term “computer readable medium” as used herein refers to any mediumthat participates in providing instructions to the processor 1203 forexecution. A computer readable medium may take many forms, including butnot limited to, non-volatile media, volatile media, and transmissionmedia. Non-volatile media includes, for example, optical, magneticdisks, and magneto-optical disks, such as the hard disk 1207 or theremovable media drive 1208. Volatile media includes dynamic memory, suchas the main memory 1204. Transmission media includes coaxial cables,copper wire and fiber optics, including the wires that make up the bus1202. Transmission media also may also take the form of acoustic orlight waves, such as those generated during radio wave and infrared datacommunications.

Various forms of computer readable media may be involved in carrying outone or more sequences of one or more instructions to processor 1203 forexecution. For example, the instructions may initially be carried on amagnetic disk of a remote computer. The remote computer can load theinstructions for implementing all or a portion of the method of videocoding or decoding remotely into a dynamic memory and send theinstructions over a telephone line using a modem. A modem local to thecomputer system 1201 may receive the data on the telephone line and usean infrared transmitter to convert the data to an infrared signal. Aninfrared detector coupled to the bus 1202 can receive the data carriedin the infrared signal and place the data on the bus 1202. The bus 1202carries the data to the main memory 1204, from which the processor 1203retrieves and executes the instructions. The instructions received bythe main memory 1204 may optionally be stored on storage device 1207 or1208 either before or after execution by processor 1203.

The computer system 1201 also includes a communication interface 1213coupled to the bus 1202. The communication interface 1213 provides atwo-way data communication coupling to a network link 1214 that isconnected to, for example, a local area network (LAN) 1215, or toanother communications network 1216 such as the Internet. For example,the communication interface 1213 may be a network interface card toattach to any packet switched LAN. As another example, the communicationinterface 1213 may be an asymmetrical digital subscriber line (ADSL)card, an integrated services digital network (ISDN) card or a modem toprovide a data communication connection to a corresponding type ofcommunications line. Wireless links may also be implemented. In any suchimplementation, the communication interface 1213 sends and receiveselectrical, electromagnetic or optical signals that carry digital datastreams representing various types of information. For example, thecommunication interface 1213 can be used to output coded moving picturesresulting from the method of video coding, or to input encoded movingpictures to process them with the method for decoding.

The network link 1214 typically provides data communication through oneor more networks to other data devices. For example, the network link1214 may provide a connection to another computer through a localnetwork 1215 (e.g., a LAN) or through equipment operated by a serviceprovider, which provides communication services through a communicationsnetwork 1216. It is possible to deliver coded video data that was codedby the method of the present invention to external devices, in an MPEG*or H.26* standard, for example a mobile device 1217, or other terminaldevices connected to network 1216. Due to the data reduction byencoding, the required bandwidth could be reduced. The local network1214 and the communications network 1216 use, for example, electrical,electromagnetic, or optical signals that carry digital data streams, andthe associated physical layer (e.g., CAT 5 cable, coaxial cable, opticalfiber, etc). The signals through the various networks and the signals onthe network link 1214 and through the communication interface 1213,which carry the digital data to and from the computer system 1201 maybeimplemented in baseband signals, or carrier wave based signals. Thebaseband signals convey the digital data as unmodulated electricalpulses that are descriptive of a stream of digital data bits, where theterm “bits” is to be construed broadly to mean symbol, where each symbolconveys at least one or more information bits. The digital data may alsobe used to modulate a carrier wave, such as with amplitude, phase and/orfrequency shift keyed signals that are propagated over a conductivemedia, or transmitted as electromagnetic waves through a propagationmedium. Thus, the digital data may be sent as unmodulated baseband datathrough a “wired” communication channel and/or sent within apredetermined frequency band, different than baseband, by modulating acarrier wave. The computer system 1201 can transmit and receive data,including program code, through the network(s) 1215 and 1216, thenetwork link 1214 and the communication interface 1213. Moreover, thenetwork link 1214 may provide a connection through a LAN 1215 to amobile device 1217 such as a personal digital assistant (PDA) laptopcomputer, or cellular telephone.

The present invention is useful to design integer transforms for imageand video coding with improved coding efficiency and lower complexitycompared to previously used integer transforms.

Although aspects of the invention have been described with reference toparticular embodiments, it is to be understood that these embodimentsare merely illustrative of the principles and applications of thepresent invention. It is therefore to be understood that numerousmodifications may be made to the illustrative embodiment and that otherarrangements may be devised without departing from the spirit and scopeof the present invention as defined by the appended claims.

The invention claimed is:
 1. A video decoding method comprising: inversetransforming with a processing device a first block of transformcoefficients by a transform matrix having a plurality of basis vectorsinto a corresponding second block of residual pixel values for a videoimage, wherein the transform matrix consists of integer elements, hasbasis vectors derived from one of a DCT and a KLT, has a DC vector, notall of the basis vectors being orthogonal but all of the basis vectorsbeing within a predetermined deviation of the DC vector, norms of thebasis vectors having respective deviations with respect to said DCvector less than 0.0015, and at least one norm-deviation being non-zero,having a maximum element deviation from said one of a DCT and a KLT lessthan or equal to 0.05, and at least one element deviation beingnon-zero, and each integer element being represented in one byte.
 2. Themethod of claim 1, wherein any inner product between two of the basisvectors is less than 0.003.
 3. The method according to claim 1, whereinsaid transform matrix is a 4×4 transform matrix and having basis vectorsdefined as: {64, 64, 64, 64,} {83, 36, -36, -83,} {64, -64, -64, 64,}{36, -83, 83, -36,}.
 4. The method according to claim 1, wherein saidtransform matrix is a 8×8 transform matrix and having basis vectorsdefined as: {64, 64, 64, 64, 64, 64, 64, 64,} {89, 75, 50,18,-18,-50,-75,-89,} {83, 36,-36,-83,-83,-36, 36, 83,} {75,-18,-89,-50,50, 89, 18,-75,} {64,-64,-64, 64, 64,-64,-64, 64,} {50,-89, 18,75,-75,-18, 89,-50,} {36,-83, 83,-36,-36, 83,-83, 36,} {18,-50, 75,-89,89,-75, 50,-18,}.
 5. The method according to claim 1, wherein saidtransform matrix is a 16×16 transform matrix and having basis vectorsdefined as: {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64,} {90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,}{89, 75, 50, 18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89,} {87,57, 9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43,-9,-57,-87,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {80,9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70,-9,-80,}{75,-18,-89,-50, 50, 89, 18,-75,-75, 18, 89, 50,-50,-89,-18, 75,}{70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90,-9, 87, 43,-70,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{57,-80,-25, 90,-9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,} {50,-89,18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89, 50,} {43,-90, 57,25,-87, 70, 9,-80, 80,-9,-70, 87,-25,-57, 90,-43,} {36,-83, 83,-36,-36,83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {25,-70, 90,-80, 43, 9,-57,87,-87, 57,-9,-43, 80,-90, 70,-25,} {18,-50, 75,-89, 89,-75, 50,-18,-18,50,-75, 89,-89, 75,-50, 18,} {9,-25, 43,-57, 70,-80, 87,-90, 90,-87,80,-70, 57,-43, 25,-9,}.
 6. The method according to claim 1, whereinsaid transform matrix is a 32×32 transform matrix and having basisvectors defined as: {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64,} {90, 90, 88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13,4,-4,-13,-22,-31,-38,-46,-54,-61, -67,-73,-78,-82,-85,-88,-90,-90,} {90,87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90,-87,-80,-70,-57,-43,-25, -9, 9, 25,43, 57, 70, 80, 87, 90,} {90, 82, 67, 46, 22,-4,-31,-54,-73,-85,-90,-88,-78,-61,-38,-13, 13, 38, 61, 78, 88, 90, 85,73, 54, 31, 4,-22,-46,-67,-82,-90,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75, -89,-89,-75,-50,-18, 18, 50, 75, 89,} {88, 67,31,-13,-54,-82,-90,-78,-46, -4, 38, 73, 90, 85, 61,22,-22,-61,-85,-90,-73,-38, 4, 46, 78, 90, 82, 54, 13,-31,-67,-88,} {87,57, 9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43, -9,-57,-87,-87,-57, -9,43, 80, 90, 70, 25, -25,-70,-90,-80,-43, 9, 57, 87,} {85,46,-13,-67,-90,-73,-22, 38, 82, 88, 54, -4,-61,-90,-78,-31, 31, 78, 90,61, 4,-54,-88,-82, -38, 22, 73, 90, 67, 13,-46,-85,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {82,22,-54,-90,-61, 13, 78, 85, 31,-46,-90,-67, 4, 73, 88, 38,-38,-88,-73,-4, 67, 90, 46,-31,-85,-78,-13, 61, 90, 54,-22,-82,} {80, 9,-70,-87,-25,57, 90, 43,-43,-90,-57, 25, 87, 70, -9,-80,-80,-9, 70, 87,25,-57,-90,-43, 43, 90, 57,-25,-87,-70, 9, 80,} {78, -4,-82,-73, 13, 85,67,-22,-88,-61, 31, 90, 54,-38,-90,-46, 46, 90, 38,-54,-90,-31, 61, 88,22,-67,-85,-13, 73, 82, 4,-78,} {75,-18,-89,-50, 50, 89, 18,-75,-75, 18,89, 50,-50,-89,-18, 75, 75,-18,-89,-50, 50, 89, 18, -75,-75, 18, 89,50,-50,-89,-18, 75,} {73,-31,-90,-22, 78, 67,-38,-90,-13, 82,61,-46,-88,-4, 85, 54,-54,-85, 4, 88, 46,-61,-82, 13, 90, 38,-67,-78,22, 90, 31,-73,} {70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90,-9, 87,43,-70,-70, 43, 87, -9,-90,-25, 80, 57, -57,-80, 25, 90, 9,-87,-43, 70,}{67,-54,-78, 38, 85,-22,-90, 4, 90, 13,-88,-31, 82, 46,-73,-61, 61,73,-46,-82, 31, 88,-13,-90, -4, 90, 22,-85,-38, 78, 54,-67,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{61,-73,-46, 82, 31,-88,-13, 90,-4,-90, 22, 85,-38,-78, 54, 67,-67,-54,78, 38,-85,-22, 90, 4, -90, 13, 88,-31,-82, 46, 73,-61,} {57,-80,-25,90, -9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,-57, 80, 25,-90, 9,87,-43,-70, 70, 43,-87,-9, 90,-25,-80, 57,} {54,-85, -4, 88,-46,-61, 82,13,-90, 38, 67,-78,-22, 90,-31,-73, 73, 31,-90, 22, 78,-67,-38, 90,-13,-82, 61, 46,-88, 4, 85,-54,} {50,-89, 18, 75,-75,-18, 89,-50,-50,89,-18,-75, 75, 18,-89, 50, 50,-89, 18, 75,-75,-18, 89, -50,-50,89,-18,-75, 75, 18,-89, 50,} {46,-90, 38, 54,-90, 31, 61,-88, 22,67,-85, 13, 73,-82, 4, 78,-78,-4, 82,-73,-13, 85,-67,-22, 88,-61,-31,90,-54,-38, 90,-46,} {43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70,87,-25,-57, 90,-43,-43, 90,-57,-25, 87,-70, -9, 80, -80, 9, 70,-87, 25,57,-90, 43,} {38,-88, 73, -4,-67, 90,-46,-31, 85,-78, 13, 61,-90, 54,22,-82, 82,-22,-54, 90,-61,-13, 78,-85, 31, 46,-90, 67, 4,-73, 88,-38,}{36,-83, 83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {31,-78,90,-61, 4, 54,-88, 82,-38,-22, 73,-90, 67,-13,-46, 85,-85, 46, 13,-67,90,-73, 22, 38, -82, 88,-54, -4, 61,-90, 78,-31,} {25,-70, 90,-80, 43,9,-57, 87,-87, 57, -9,-43, 80,-90, 70,-25,-25, 70,-90, 80,-43, -9,57,-87, 87,-57, 9, 43,-80, 90,-70, 25,} {22,-61, 85,-90, 73,-38, -4,46,-78, 90,-82, 54,-13,-31, 67,-88, 88,-67, 31, 13,-54, 82,-90, 78, -46,4, 38,-73, 90,-85, 61,-22,} {18,-50, 75,-89, 89,-75, 50,-18,-18, 50,-75,89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75, 50, -18,-18, 50,-75, 89,-89,75,-50, 18,} {13,-38, 61,-78, 88,-90, 85,-73, 54,-31, 4, 22,-46, 67,-82,90,-90, 82,-67, 46,-22, -4, 31,-54, 73,-85, 90,-88, 78,-61, 38,-13,}{9,-25, 43,-57, 70,-80, 87,-90, 90,-87, 80,-70, 57,-43, 25, -9, -9,25,-43, 57,-70, 80,-87, 90, -90, 87,-80, 70,-57, 43,-25, 9,} {4,-13,22,-31, 38,-46, 54,-61, 67,-73, 78,-82, 85,-88, 90,-90, 90,-90, 88,-85,82,-78, 73,-67, 61,-54, 46,-38, 31,-22, 13, -4,}.
 7. The methodaccording to claim 1, wherein said transform matrix is a 64×64 transformmatrix and having basis vectors defined as: {64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64,} {91, 90, 90, 89, 88, 87, 86, 84, 83, 81, 79, 76, 74, 71,69, 66, 62, 59, 56, 52, 48, 45, 41, 37, 33, 28, 24, 20, 16, 11, 7, 2,-2, -7,-11,-16,-20,-24,-28,-33,-37,-41,-45,-48,-52,-56,-59,-62,-66,-69,-71,-74,-76,-79,-81,-83,-84,-86,-87,-88,-89,-90,-90,-91,} {90, 90,88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4,-4,-13,-22,-31,-38,-46,-54,-61,-67,-73,-78,-82,-85,-88,-90,-90,-90,-90,-88,-85,-82,-78,-73,-67,-61,-54,-46,-38,-31,-22,-13,-4, 4, 13, 22, 31, 38, 46, 54, 61, 67, 73, 78, 82, 85, 88, 90, 90,} {90,88, 84, 79, 71, 62, 52, 41, 28, 16,2,-11,-24,-37,-48,-59,-69,-76,-83,-87,-90,-91,-89,-86,-81,-74,-66,-56,-45,-33,-20,-7, 7, 20, 33, 45, 56, 66, 74, 81, 86, 89,91, 90, 87, 83, 76, 69, 59, 48, 37, 24,11,-2,-16,-28,-41,-52,-62,-71,-79,-84,-88,-90,} {90, 87, 80, 70, 57, 43,25, 9, -9,-25,-43,-57,-70,-80,-87,-90,-90,-87,-80,-70,-57,-43,-25,-9, 9,25, 43, 57, 70, 80, 87, 90, 90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90, -87,-80,-70,-57,-43,-25,-9, 9, 25,43, 57, 70, 80, 87, 90,} {90, 84, 74, 59, 41,20,-2,-24,-45,-62,-76,-86,-90,-89,-83,-71,-56,-37,-16, 7, 28, 48, 66,79, 87, 91, 88, 81, 69, 52, 33,11,-11,-33,-52,-69,-81,-88,-91,-87,-79,-66,-48,-28, -7, 16, 37, 56, 71,83, 89, 90, 86, 76, 62, 45, 24, 2,-20,-41,-59,-74,-84,-90,} {90, 82, 67,46, 22,-4,-31,-54,-73,-85,-90,-88,-78,-61,-38,-13, 13, 38, 61, 78, 88,90, 85, 73, 54, 31, 4,-22,-46,-67,-82,-90,-90,-82,-67,-46,-22, 4, 31,54, 73, 85, 90, 88, 78, 61, 38, 13,-13,-38,-61,-78,-88,-90,-85,-73,-54,-31, -4, 22, 46, 67, 82, 90,} {89, 79,59, 33, 2,-28,-56,-76,-88,-90,-81,-62,-37, -7, 24, 52, 74, 87, 90, 83,66, 41, 11,-20, -48,-71,-86,-91,-84,-69,-45,-16, 16, 45, 69, 84, 91, 86,71, 48, 20,-11,-41,-66,-83,-90,-87,-74, -52,-24, 7, 37, 62, 81, 90, 88,76, 56, 28,-2,-33,-59,-79,-89,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75, -89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89,} {88, 71, 41,2,-37,-69,-87,-89,-74,-45, -7, 33, 66, 86, 90, 76, 48,11,-28,-62,-84,-90,-79,-52, -16, 24, 59, 83, 91, 81, 56,20,-20,-56,-81,-91,-83,-59,-24, 16, 52, 79, 90, 84, 62, 28,-11,-48,-76,-90,-86,-66,-33, 7, 45, 74, 89, 87, 69, 37, -2,-41,-71,-88,} {88,67, 31,-13,-54,-82,-90,-78,-46, -4, 38, 73, 90, 85, 61,22,-22,-61,-85,-90,-73,-38, 4, 46, 78, 90, 82, 54,13,-31,-67,-88,-88,-67,-31, 13, 54, 82, 90, 78, 46,4,-38,-73,-90,-85,-61,-22, 22, 61, 85, 90, 73, 38,-4,-46,-78,-90,-82,-54,-13, 31, 67, 88,} {87, 62,20,-28,-69,-89,-84,-56,-11, 37, 74, 90, 81, 48, 2,-45,-79,-91,-76,-41,7, 52, 83, 90, 71, 33,-16,-59,-86,-88,-66,-24, 24, 66, 88, 86, 59,16,-33,-71,-90,-83,-52, -7, 41, 76, 91, 79, 45, -2,-48,-81,-90,-74,-37,11, 56, 84, 89, 69, 28,-20,-62,-87,} {87, 57, 9,-43,-80,-90,-70,-25, 25,70, 90, 80, 43, -9,-57,-87,-87,-57, -9, 43, 80, 90, 70, 25,-25,-70,-90,-80,-43, 9, 57, 87, 87, 57, 9,-43,-80,-90,-70,-25, 25, 70,90, 80, 43, -9,-57,-87,-87, -57,-9, 43, 80, 90, 70,25,-25,-70,-90,-80,-43, 9, 57, 87,} {86, 52, -2,-56,-87,-84,-48, 7, 59,88, 83, 45,-11,-62,-89,-81,-41, 16, 66, 90, 79, 37,-20,-69, -90,-76,-33,24, 71, 91, 74, 28,-28,-74,-91,-71,-24, 33, 76, 90, 69,20,-37,-79,-90,-66,-16, 41, 81, 89, 62, 11,-45,-83,-88,-59, -7, 48, 84,87, 56, 2,-52,-86,} {85, 46,-13,-67,-90,-73,-22, 38, 82, 88, 54,-4,-61,-90,-78,-31, 31, 78, 90, 61, 4,-54,-88,-82, -38, 22, 73, 90, 67,13,-46,-85,-85,-46, 13, 67, 90, 73, 22,-38,-82,-88,-54, 4, 61, 90, 78,31, -31,-78,-90,-61, -4, 54, 88, 82, 38,-22,-73,-90,-67,-13, 46, 85,}{84, 41,-24,-76,-89,-56, 7, 66, 91, 69, 11,-52,-88,-79,-28, 37, 83, 86,45,-20,-74,-90,-59, 2, 62, 90, 71, 16,-48,-87,-81,-33, 33, 81, 87,48,-16,-71,-90,-62,-2, 59, 90, 74, 20,-45,-86,-83, -37, 28, 79, 88,52,-11,-69,-91,-66, -7, 56, 89, 76, 24,-41,-84,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {83,28,-45,-88,-74,-11, 59, 91, 62, -7,-71,-89,-48, 24, 81, 84,33,-41,-87,-76,-16, 56, 90, 66, --2,-69,-90,-52, 20, 79, 86,37,-37,-86,-79,-20, 52, 90, 69, 2,-66,-90,-56, 16, 76, 87, 41,-33,-84,-81,-24, 48, 89, 71, 7,-62,-91,-59, 11, 74, 88, 45,-28,-83,} {82,22,-54,-90,-61, 13, 78, 85, 31,-46,-90,-67, 4, 73, 88, 38,-38,-88,-73,-4, 67, 90, 46,-31, -85,-78,-13, 61, 90, 54,-22,-82,-82,-22, 54, 90,61,-13,-78,-85,-31, 46, 90, 67, -4,-73,-88,-38, 38, 88, 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46,-82,-31, 88, 13,-90, 4, 90,-22,-85, 38, 78,-54,-67,67, 54,-78,-38, 85, 22,-90, -4, 90,-13,-88, 31, 82,-46,-73, 61,}{59,-76,-37, 87, 11,-91, 16, 86,-41,-74, 62, 56,-79,-33, 88, 7,-90, 20,84,-45,-71, 66, 52,-81, -28, 89, 2,-90, 24, 83,-48,-69, 69, 48,-83,-24,90, -2,-89, 28, 81,-52,-66, 71, 45,-84,-20, 90, -7, -88, 33, 79,-56,-62,74, 41,-86,-16, 91,-11,-87, 37, 76,-59,} {57,-80,-25, 90, -9,-87, 43,70,-70,-43, 87, 9,-90, 25, 80,-57,-57, 80, 25,-90, 9, 87,-43,-70, 70,43,-87, -9, 90,-25,-80, 57, 57,-80,-25, 90, -9,-87, 43, 70,-70,-43, 87,9,-90, 25, 80,-57,-57, 80, 25,-90, 9, 87,-43,-70, 70, 43,-87, -9,90,-25,-80, 57,} {56,-83,-16, 90,-28,-76, 66, 45,-87,-2, 88,-41,-69, 74,33,-90, 11, 84,-52,-59, 81, 20,-91, 24, 79,-62,-48, 86, 7,-89, 37,71,-71,-37, 89,-7,-86, 48, 62,-79,-24, 91,-20,-81, 59, 52,-84,-11,90,-33,-74, 69, 41,-88, 2, 87,-45,-66, 76, 28,-90, 16, 83,-56,} {54,-85,-4, 88,-46,-61, 82, 13,-90, 38, 67,-78,-22, 90,-31,-73, 73, 31,-90, 22,78,-67,-38, 90, -13,-82, 61, 46,-88, 4, 85,-54,-54, 85, 4,-88, 46,61,-82,-13, 90,-38,-67, 78, 22,-90, 31, 73,-73, -31, 90,-22,-78, 67,38,-90, 13, 82,-61,-46, 88, -4,-85, 54,} {52,-87, 7, 83,-62,-41,90,-20,-76, 71, 28,-91, 33, 69,-79,-16, 89,-45,-59, 84, 2,-86, 56, 48,-88, 11, 81,-66,-37, 90,-24,-74, 74, 24,-90, 37, 66,-81,-11, 88,-48,-56,86, -2,-84, 59, 45,-89, 16, 79,-69,-33, 91,-28,-71, 76, 20,-90, 41,62,-83, -7, 87,-52,} {50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75,75, 18,-89, 50, 50,-89, 18, 75,-75,-18, 89, -50,-50, 89,-18,-75, 75,18,-89, 50, 50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89,50, 50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89, 50,}{48,-90, 28, 66,-84, 7, 79,-74,-16, 87,-59,-37, 91,-41,-56, 88,-20,-71,81, 2,-83, 69, 24,-89, 52, 45,-90, 33, 62,-86, 11, 76,-76,-11,86,-62,-33, 90,-45,-52, 89,-24,-69, 83,-2,-81, 71, 20, -88, 56, 41,-91,37, 59,-87, 16, 74,-79, -7, 84,-66,-28, 90,-48,} {46,-90, 38, 54,-90,31, 61,-88, 22, 67,-85, 13, 73,-82, 4, 78,-78,-4, 82,-73,-13,85,-67,-22, 88,-61,-31, 90,-54,-38, 90,-46,-46, 90,-38,-54, 90,-31,-61,88,-22,-67, 85,-13,-73, 82, -4,-78, 78, 4,-82, 73, 13,-85, 67, 22,-88,61, 31,-90, 54, 38,-90, 46,} {45,-91, 48, 41,-90, 52, 37,-90, 56,33,-89, 59, 28,-88, 62, 24,-87, 66, 20,-86, 69, 16,-84, 71, 11,-83, 74,7,-81, 76, 2,-79, 79, -2,-76, 81, -7,-74, 83,-11,-71, 84,-16,-69,86,-20,-66, 87,-24, -62, 88,-28,-59, 89,-33,-56, 90,-37,-52, 90,-41,-48,91,-45,} {43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70, 87,-25,-57,90,-43,-43, 90,-57,-25, 87,-70, -9, 80, -80, 9, 70,-87, 25, 57,-90, 43,43,-90, 57, 25,-87, 70, 9,-80, 80,-9,-70, 87,-25,-57, 90,-43,-43,90,-57,-25, 87,-70,-9, 80,-80, 9, 70,-87, 25, 57,-90, 43,} {41,-89, 66,11,-79, 83,-20,-59, 90,-48,-33, 87,-71, -2, 74,-86, 28, 52,-91, 56,24,-84, 76, -7, -69, 88,-37,-45, 90,-62,-16, 81,-81, 16, 62,-90, 45,37,-88, 69, 7,-76, 84,-24,-56, 91,-52,-28, 86,-74, 2, 71,-87, 33,48,-90, 59, 20,-83, 79,-11,-66, 89,-41,} {38,-88, 73,-4,-67, 90,-46,-31,85,-78, 13, 61,-90, 54, 22,-82, 82,-22,-54, 90,-61,-13, 78,-85, 31,46,-90, 67, 4,-73, 88,-38,-38, 88,-73, 4, 67,-90, 46, 31,-85,78,-13,-61, 90,-54,-22, 82,-82, 22, 54,-90, 61, 13,-78, 85,-31,-46,90,-67, -4, 73,-88, 38,} {37,-86, 79,-20,-52, 90,-69, 2, 66,-90, 56,16,-76, 87,-41,-33, 84,-81, 24, 48,-89, 71, -7,-62, 91,-59,-11, 74,-88,45, 28,-83, 83,-28,-45, 88,-74, 11, 59,-91, 62, 7,-71, 89,-48,-24,81,-84, 33, 41,-87, 76,-16,-56, 90,-66, -2, 69,-90, 52, 20,-79, 86,-37,}{36,-83, 83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {33,-81,87,-48,-16, 71,-90, 62, -2,-59, 90,-74, 20, 45,-86, 83,-37,-28, 79,-88,52, 11,-69, 91, -66, 7, 56,-89, 76,-24,-41, 84,-84, 41, 24,-76, 89,-56,-7, 66,-91, 69,-11,-52, 88,-79, 28, 37,-83, 86,-45,-20, 74,-90, 59,2,-62, 90,-71, 16, 48,-87, 81,-33,} {31,-78, 90,-61, 4, 54,-88,82,-38,-22, 73,-90, 67,-13,-46, 85,-85, 46, 13,-67, 90,-73, 22, 38, -82,88,-54, -4, 61,-90, 78,-31,-31, 78,-90, 61, -4,-54, 88,-82, 38, 22,-73,90,-67, 13, 46,-85, 85,-46,-13, 67,-90, 73,-22,-38, 82,-88, 54, 4,-61,90,-78, 31,} {28,-74, 91,-71, 24, 33,-76, 90,-69, 20, 37,-79, 90,-66,16, 41,-81, 89,-62, 11, 45,-83, 88,-59, 7, 48,-84, 87,-56, 2, 52,-86,86,-52, -2, 56,-87, 84,-48,-7, 59,-88, 83,-45,-11, 62,-89, 81,-41, -16,66,-90, 79,-37,-20, 69,-90, 76,-33,-24, 71,-91, 74,-28,} {25,-70,90,-80, 43, 9,-57, 87,-87, 57,-9,-43, 80,-90, 70,-25,-25, 70,-90,80,-43, -9, 57,-87, 87,-57, 9, 43,-80, 90,-70, 25, 25,-70, 90,-80, 43,9,-57, 87,-87, 57, -9,-43, 80,-90, 70,-25,-25, 70,-90, 80,-43, -9,57,-87, 87,-57, 9, 43,-80, 90,-70, 25,} {24,-66, 88,-86, 59,-16,-33,71,-90, 83,-52, 7, 41,-76, 91,-79, 45, 2,-48, 81,-90, 74,-37,-11,56,-84, 89,-69, 28, 20,-62, 87,-87, 62,-20,-28, 69,-89, 84,-56, 11,37,-74, 90,-81, 48, -2,-45, 79,-91, 76,-41,-7, 52,-83, 90,-71, 33,16,-59, 86,-88, 66,-24,} {22,-61, 85,-90, 73,-38, -4, 46,-78, 90,-82,54,-13,-31, 67,-88, 88,-67, 31, 13,-54, 82,-90, 78, -46, 4, 38,-73,90,-85, 61,-22,-22, 61,-85, 90,-73, 38, 4,-46, 78,-90, 82,-54, 13,31,-67, 88,-88, 67,-31,-13, 54,-82, 90,-78, 46, -4,-38, 73,-90, 85,-61,22,} {20,-56, 81,-91, 83,-59, 24, 16,-52, 79,-90, 84,-62, 28, 11,-48,76,-90, 86,-66, 33, 7,-45, 74, -89, 87,-69, 37, 2,-41, 71,-88, 88,-71,41, -2,-37, 69,-87, 89,-74, 45,-7,-33, 66,-86, 90,-76, 48, -11,-28,62,-84, 90,-79, 52,-16,-24, 59,-83, 91,-81, 56,-20,} {18,-50, 75,-89,89,-75, 50,-18,-18, 50,-75, 89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75,50, -18,-18, 50,-75, 89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75,50,-18,-18, 50,-75, 89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75,50,-18,-18, 50,-75, 89,-89, 75,-50, 18,} {16,-45, 69,-84, 91,-86,71,-48, 20, 11,-41, 66,-83, 90,-87, 74,-52, 24, 7,-37, 62,-81, 90,-88,76,-56, 28, 2,-33, 59,-79, 89,-89, 79,-59, 33, -2,-28, 56,-76, 88,-90,81,-62, 37, -7,-24, 52,-74, 87,-90, 83,-66, 41,-11,-20, 48,-71, 86,-91,84,-69, 45,-16,} {13,-38, 61,-78, 88,-90, 85,-73, 54,-31, 4, 22,-46,67,-82, 90,-90, 82,-67, 46,-22, -4, 31,-54, 73,-85, 90,-88, 78,-61,38,-13,-13, 38,-61, 78,-88, 90,-85, 73,-54, 31, -4,-22, 46,-67, 82,-90,90,-82, 67,-46, 22, 4,-31, 54,-73, 85,-90, 88,-78, 61,-38, 13,} {11,-33,52,-69, 81,-88, 91,-87, 79,-66, 48,-28, 7, 16,-37, 56,-71, 83,-89,90,-86, 76,-62, 45, -24, 2, 20,-41, 59,-74, 84,-90, 90,-84, 74,-59,41,-20,-2, 24,-45, 62,-76, 86,-90, 89,-83, 71,-56, 37,-16, -7, 28,-48,66,-79, 87,-91, 88,-81, 69,-52, 33,-11,} {9,-25, 43,-57, 70,-80, 87,-90,90,-87, 80,-70, 57,-43, 25,-9, -9, 25,-43, 57,-70, 80,-87, 90, -90,87,-80, 70,-57, 43,-25, 9, 9,-25, 43,-57, 70,-80, 87,-90, 90,-87,80,-70, 57,-43, 25, -9, -9, 25,-43, 57,-70, 80,-87, 90,-90, 87,-80,70,-57, 43,-25, 9,} {7,-20, 33,-45, 56,-66, 74,-81, 86,-89, 91,-90,87,-83, 76,-69, 59,-48, 37,-24, 11, 2,-16, 28, -41, 52,-62, 71,-79,84,-88, 90,-90, 88,-84, 79,-71, 62,-52, 41,-28, 16, -2,-11, 24,-37,48,-59, 69,-76, 83,-87, 90,-91, 89,-86, 81,-74, 66,-56, 45,-33, 20, -7,}{4,-13, 22,-31, 38,-46, 54,-61, 67,-73, 78,-82, 85,-88, 90,-90, 90,-90,88,-85, 82,-78, 73,-67, 61,-54, 46,-38, 31,-22, 13, -4, -4, 13,-22,31,-38, 46,-54, 61,-67, 73,-78, 82,-85, 88,-90, 90, -90, 90,-88, 85,-82,78,-73, 67,-61, 54,-46, 38,-31, 22,-13, 4,} {2, -7, 11,-16, 20,-24,28,-33, 37,-41, 45,-48, 52,-56, 59,-62, 66,-69, 71,-74, 76,-79, 81,-83,84,-86, 87,-88, 89,-90, 90,-91, 91,-90, 90,-89, 88,-87, 86,-84, 83,-81,79,-76, 74,-71, 69,-66, 62,-59, 56,-52, 48,-45, 41,-37, 33,-28, 24,-20,16,-11, 7, -2,}.
 8. A video decoder comprising: a non-transitorycomputer readable storage device having stored therein a transformmatrix having a plurality of basis vectors; and a processing deviceconfigured to inverse transform with a first block of transformcoefficients by the transform matrix into a corresponding second blockof residual pixel values for a video image, wherein the transform matrixconsists of integer elements, has basis vectors derived from one of aDCT and a KLT, has a DC vector, not all of the basis vectors beingorthogonal but all of the basis vectors being within a predetermineddeviation of the DC vector, norms of the basis vectors having respectivedeviations with respect to said DC vector less than 0.0015, and at leastone norm-deviation being non-zero, having a maximum element deviationfrom said one of a DCT and a KLT less than or equal to 0.05, and atleast one element deviation being non-zero, and each integer elementbeing represented in one byte.
 9. The video decoder of claim 8, whereinany inner product between two of the basis vectors is less than 0.003.10. The video decoder of claim 8, wherein said transform matrix is a 4×4transform matrix and having basis vectors defined as: {64, 64, 64, 64,}{83, 36,-36,-83,} {64,-64,-64, 64,} {36,-83, 83,-36,}.
 11. The videodecoder of claim 8, wherein said transform matrix is a 8×8 transformmatrix and having basis vectors defined as: {64, 64, 64, 64, 64, 64, 64,64,} {89, 75, 50, 18,-18,-50,-75,-89,} {83, 36,-36,-83,-83,-36, 36, 83,}{75,-18,-89,-50, 50, 89, 18,-75,} {64,-64,-64, 64, 64,-64,-64, 64,}{50,-89, 18, 75,-75,-18, 89,-50,} {36,-83, 83,-36,-36, 83,-83, 36,}{18,-50, 75,-89, 89,-75, 50,-18,}.
 12. The video decoder of claim 8,wherein said transform matrix is a 16×16 transform matrix and havingbasis vectors defined as: {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64,} {90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89,} {87, 57,9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43, -9,-57,-87,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {80,9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70, -9,-80,}{75,-18,-89,-50, 50, 89, 18,-75,-75, 18, 89, 50,-50,-89,-18, 75,}{70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90,-9, 87, 43,-70,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{57,-80,-25, 90, -9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,}{50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89, 50,}{43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70, 87,-25,-57, 90,-43,}{36,-83, 83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,}{25,-70, 90,-80, 43, 9,-57, 87,-87, 57, -9,-43, 80,-90, 70,-25,}{18,-50, 75,-89, 89,-75, 50,-18,-18, 50,-75, 89,-89, 75,-50, 18,}{9,-25, 43,-57, 70,-80, 87,-90, 90,-87, 80,-70, 57,-43, 25, -9,}. 13.The video decoder of claim 8, wherein wherein said transform matrix is a32×32 transform matrix and having basis vectors defined as: {64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,} {90, 90, 88, 85, 82, 78,73, 67, 61, 54, 46, 38, 31, 22, 13, 4, -4,-13,-22,-31,-38,-46,-54,-61,-67,-73,-78,-82,-85,-88,-90,-90,} {90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90,-87,-80,-70,-57,-43,-25, -9, 9, 25,43, 57, 70, 80, 87, 90,} {90, 82, 67, 46, 22,-4,-31,-54,-73,-85,-90,-88,-78,-61,-38,-13, 13, 38, 61, 78, 88, 90, 85,73, 54, 31, 4,-22,-46,-67,-82,-90,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75, -89,-89,-75,-50,-18, 18, 50, 75, 89,} {88, 67,31,-13,-54,-82,-90,-78,-46, -4, 38, 73, 90, 85, 61,22,-22,-61,-85,-90,-73,-38, 4, 46, 78, 90, 82, 54, 13,-31,-67,-88,} {87,57, 9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43, -9,-57,-87,-87,-57, -9,43, 80, 90, 70, 25, -25,-70,-90,-80,-43, 9, 57, 87,} {85,46,-13,-67,-90,-73,-22, 38, 82, 88, 54, -4,-61,-90,-78,-31, 31, 78, 90,61, 4,-54,-88,-82, -38, 22, 73, 90, 67, 13,-46,-85,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {82,22,-54,-90,-61, 13, 78, 85, 31,-46,-90,-67, 4, 73, 88, 38,-38,-88,-73,-4, 67, 90, 46,-31, -85,-78,-13, 61, 90, 54,-22,-82,} {80,9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70, -9,-80,-80, -9, 70,87, 25,-57,-90,-43, 43, 90, 57,-25,-87,-70, 9, 80,} {78, -4,-82,-73, 13,85, 67,-22,-88,-61, 31, 90, 54,-38,-90,-46, 46, 90, 38,-54,-90,-31, 61,88, 22,-67,-85,-13, 73, 82, 4,-78,} {75,-18,-89,-50, 50, 89, 18,-75,-75,18, 89, 50,-50,-89,-18, 75, 75,-18,-89,-50, 50, 89, 18, -75,-75, 18, 89,50,-50,-89,-18, 75,} {73,-31,-90,-22, 78, 67,-38,-90,-13, 82,61,-46,-88, -4, 85, 54,-54,-85, 4, 88, 46,-61,-82, 13, 90, 38,-67,-78,22, 90, 31,-73,} {70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90, -9, 87,43,-70,-70, 43, 87, -9,-90,-25, 80, 57, -57,-80, 25, 90, 9,-87,-43, 70,}{67,-54,-78, 38, 85,-22,-90, 4, 90, 13,-88,-31, 82, 46,-73,-61, 61,73,-46,-82, 31, 88,-13,-90, -4, 90, 22,-85,-38, 78, 54,-67,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{61,-73,-46, 82, 31,-88,-13, 90, -4,-90, 22, 85,-38,-78, 54, 67,-67,-54,78, 38,-85,-22, 90, 4, -90, 13, 88,-31,-82, 46, 73,-61,} {57,-80,-25,90,-9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,-57, 80, 25,-90, 9,87,-43,-70, 70, 43,-87, -9, 90,-25,-80, 57,} {54,-85, -4, 88,-46,-61,82, 13,-90, 38, 67,-78,-22, 90,-31,-73, 73, 31,-90, 22, 78,-67,-38, 90,-13,-82, 61, 46,-88, 4, 85,-54,} {50,-89, 18, 75,-75,-18, 89,-50,-50,89,-18,-75, 75, 18,-89, 50, 50,-89, 18, 75,-75,-18, 89, -50,-50,89,-18,-75, 75, 18,-89, 50,} {46,-90, 38, 54,-90, 31, 61,-88, 22,67,-85, 13, 73,-82, 4, 78,-78, -4, 82,-73,-13, 85,-67,-22, 88,-61,-31,90,-54,-38, 90,-46,} {43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70,87,-25,-57, 90,-43,-43, 90,-57,-25, 87,-70, -9, 80, -80, 9, 70,-87, 25,57,-90, 43,} {38,-88, 73,-4,-67, 90,-46,-31, 85,-78, 13, 61,-90, 54,22,-82, 82,-22,-54, 90,-61,-13, 78,-85, 31, 46,-90, 67, 4,-73, 88,-38,}{36,-83, 83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {31,-78,90,-61, 4, 54,-88, 82,-38,-22, 73,-90, 67,-13,-46, 85,-85, 46, 13,-67,90,-73, 22, 38, -82, 88,-54, -4, 61,-90, 78,-31,} {25,-70, 90,-80, 43,9,-57, 87,-87, 57, -9,-43, 80,-90, 70,-25,-25, 70,-90, 80,-43,-9,57,-87, 87,-57, 9, 43,-80, 90,-70, 25,} {22,-61, 85,-90, 73,-38, -4,46,-78, 90,-82, 54,-13,-31, 67,-88, 88,-67, 31, 13,-54, 82,-90, 78, -46,4, 38,-73, 90,-85, 61,-22,} {18,-50, 75,-89, 89,-75, 50,-18,-18, 50,-75,89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75, 50, -18,-18, 50,-75, 89,-89,75,-50, 18,} {13,-38, 61,-78, 88,-90, 85,-73, 54,-31, 4, 22,-46, 67,-82,90,-90, 82,-67, 46,-22, -4, 31,-54, 73,-85, 90,-88, 78,-61, 38,-13,}{9,-25, 43,-57, 70,-80, 87,-90, 90,-87, 80,-70, 57,-43, 25, -9, -9,25,-43, 57,-70, 80,-87, 90, -90, 87,-80, 70,-57, 43,-25, 9,} {4,-13,22,-31, 38,-46, 54,-61, 67,-73, 78,-82, 85,-88, 90,-90, 90,-90, 88,-85,82,-78, 73,-67, 61,-54, 46,-38, 31,-22, 13, -4,}.
 14. The video decoderof claim 8, wherein said transform matrix is a 64×64 transform matrixand having basis vectors defined as: {64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64,} {91, 90, 90, 89, 88, 87, 86, 84, 83, 81, 79, 76, 74, 71, 69,66, 62, 59, 56, 52, 48, 45, 41, 37, 33, 28, 24, 20, 16, 11, 7, 2, -2,-7,-11,-16,-20,-24,-28,-33,-37,-41,-45,-48,-52,-56,-59,-62,-66,-69,-71,-74,-76,-79,-81,-83,-84,-86,-87,-88,-89,-90,-90,-91,} {90, 90,88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4,-4,-13,-22,-31,-38,-46,-54,-61,-67,-73,-78,-82,-85,-88,-90,-90,-90,-90,-88,-85,-82,-78,-73,-67,-61,-54,-46,-38,-31,-22,-13,-4, 4, 13, 22, 31, 38, 46, 54, 61, 67, 73, 78, 82, 85, 88, 90, 90,} {90,88, 84, 79, 71, 62, 52, 41, 28, 16,2,-11,-24,-37,-48,-59,-69,-76,-83,-87,-90,-91,-89,-86,-81,-74,-66,-56,-45,-33,-20, -7, 7, 20, 33, 45, 56, 66, 74, 81, 86, 89,91, 90, 87, 83, 76, 69, 59, 48, 37, 24, 11,-2,-16,-28,-41,-52,-62,-71,-79,-84,-88,-90,} {90, 87, 80, 70, 57, 43,25, 9, -9,-25,-43,-57,-70,-80,-87,-90,-90,-87,-80,-70,-57,-43,-25, -9,9, 25, 43, 57, 70, 80, 87, 90, 90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90, -87,-80,-70,-57,-43,-25, -9, 9, 25,43, 57, 70, 80, 87, 90,} {90, 84, 74, 59, 41, 20,-2,-24,-45,-62,-76,-86,-90,-89,-83,-71,-56,-37,-16, 7, 28, 48, 66, 79,87, 91, 88, 81, 69, 52, 33,11,-11,-33,-52,-69,-81,-88,-91,-87,-79,-66,-48,-28, -7, 16, 37, 56, 71,83, 89, 90, 86, 76, 62, 45, 24, 2,-20,-41,-59,-74,-84,-90,} {90, 82, 67,46, 22, -4,-31,-54,-73,-85,-90,-88,-78,-61,-38,-13, 13, 38, 61, 78, 88,90, 85, 73, 54, 31, 4,-22,-46,-67,-82,-90,-90,-82,-67,-46,-22, 4, 31,54, 73, 85, 90, 88, 78, 61, 38, 13,-13,-38,-61,-78,-88,-90,-85,-73,-54,-31, -4, 22, 46, 67, 82, 90,} {89, 79,59, 33, 2,-28,-56,-76,-88,-90,-81,-62,-37, -7, 24, 52, 74, 87, 90, 83,66, 41, 11,-20, -48,-71,-86,-91,-84,-69,-45,-16, 16, 45, 69, 84, 91, 86,71, 48, 20,-11,-41,-66,-83,-90,-87,-74, -52,-24, 7, 37, 62, 81, 90, 88,76, 56, 28, -2,-33,-59,-79,-89,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75, -89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89,} {88, 71, 41,2,-37,-69,-87,-89,-74,-45, -7, 33, 66, 86, 90, 76, 48,11,-28,-62,-84,-90,-79,-52, -16, 24, 59, 83, 91, 81, 56,20,-20,-56,-81,-91,-83,-59,-24, 16, 52, 79, 90, 84, 62, 28,-11,-48,-76,-90,-86,-66,-33, 7, 45, 74, 89, 87, 69, 37, -2,-41,-71,-88,} {88,67, 31,-13,-54,-82,-90,-78,-46, -4, 38, 73, 90, 85, 61,22,-22,-61,-85,-90,-73,-38, 4, 46, 78, 90, 82, 54,13,-31,-67,-88,-88,-67,-31, 13, 54, 82, 90, 78, 46,4,-38,-73,-90,-85,-61,-22, 22, 61, 85, 90, 73, 38,-4,-46,-78,-90,-82,-54,-13, 31, 67, 88,} {87, 62,20,-28,-69,-89,-84,-56,-11, 37, 74, 90, 81, 48, 2,-45,-79,-91,-76,-41,7, 52, 83, 90, 71, 33,-16,-59,-86,-88,-66,-24, 24, 66, 88, 86, 59,16,-33,-71,-90,-83,-52, -7, 41, 76, 91, 79, 45, -2,-48,-81,-90,-74,-37,11, 56, 84, 89, 69, 28,-20,-62,-87,} {87, 57, 9,-43,-80,-90,-70,-25, 25,70, 90, 80, 43, -9,-57,-87,-87,-57, -9, 43, 80, 90, 70, 25,-25,-70,-90,-80,-43, 9, 57, 87, 87, 57, 9,-43,-80,-90,-70,-25, 25, 70,90, 80, 43, -9,-57,-87,-87, -57, -9, 43, 80, 90, 70,25,-25,-70,-90,-80,-43, 9, 57, 87,} {86, 52, -2,-56,-87,-84,-48, 7, 59,88, 83, 45,-11,-62,-89,-81,-41, 16, 66, 90, 79, 37,-20,-69, -90,-76,-33,24, 71, 91, 74, 28,-28,-74,-91,-71,-24, 33, 76, 90, 69,20,-37,-79,-90,-66,-16, 41, 81, 89, 62, 11,-45,-83,-88,-59, -7, 48, 84,87, 56, 2,-52,-86,} {85, 46,-13,-67,-90,-73,-22, 38, 82, 88, 54,-4,-61,-90,-78,-31, 31, 78, 90, 61, 4,-54,-88,-82, -38, 22, 73, 90, 67,13,-46,-85,-85,-46, 13, 67, 90, 73, 22,-38,-82,-88,-54, 4, 61, 90, 78,31, -31,-78,-90,-61, -4, 54, 88, 82, 38,-22,-73,-90,-67,-13, 46, 85,}{84, 41,-24,-76,-89,-56, 7, 66, 91, 69, 11,-52,-88,-79,-28, 37, 83, 86,45,-20,-74,-90,-59, 2, 62, 90, 71, 16,-48,-87,-81,-33, 33, 81, 87,48,-16,-71,-90,-62, -2, 59, 90, 74, 20,-45,-86,-83, -37, 28, 79, 88,52,-11,-69,-91,-66,-7, 56, 89, 76, 24,-41,-84,} {83, 36,-36,-83,-83,-36,36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,83, 36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83,} {83, 28,-45,-88,-74,-11, 59, 91, 62,-7,-71,-89,-48, 24, 81, 84, 33,-41,-87,-76,-16, 56, 90, 66,-2,-69,-90,-52, 20, 79, 86, 37,-37,-86,-79,-20, 52, 90, 69,2,-66,-90,-56, 16, 76, 87, 41,-33, -84,-81,-24, 48, 89, 71,7,-62,-91,-59, 11, 74, 88, 45,-28,-83,} {82, 22,-54,-90,-61, 13, 78, 85,31,-46,-90,-67, 4, 73, 88, 38,-38,-88,-73, -4, 67, 90, 46,-31,-85,-78,-13, 61, 90, 54,-22,-82,-82,-22, 54, 90, 61,-13,-78,-85,-31, 46,90, 67, -4,-73,-88,-38, 38, 88, 73, 4,-67,-90,-46, 31, 85, 78,13,-61,-90,-54, 22, 82,} {81, 16,-62,-90,-45, 37, 88, 69,-7,-76,-84,-24, 56, 91, 52,-28,-86,-74, -2, 71, 87, 33,-48,-90, -59, 20,83, 79, 11,-66,-89,-41, 41, 89, 66,-11,-79,-83,-20, 59, 90,48,-33,-87,-71, 2, 74, 86, 28,-52,-91,-56, 24, 84, 76, 7,-69,-88,-37,45, 90, 62,-16,-81,} {80, 9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87,70, -9,-80,-80, -9, 70, 87, 25,-57,-90,-43, 43, 90, 57,-25,-87,-70, 9,80, 80, 9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70, -9,-80,-80,-9, 70, 87, 25,-57,-90,-43, 43, 90, 57,-25,-87,-70, 9, 80,} {79,2,-76,-81, -7, 74, 83, 11,-71,-84,-16, 69, 86, 20,-66,-87,-24, 62, 88,28,-59,-89,-33, 56, 90, 37,-52,-90,-41, 48, 91, 45,-45,-91,-48, 41, 90,52,-37,-90,-56, 33, 89, 59,-28,-88,-62, 24, 87, 66,-20,-86,-69, 16, 84,71,-11,-83,-74, 7, 81, 76, -2,-79,} {78, -4,-82,-73, 13, 85,67,-22,-88,-61, 31, 90, 54,-38,-90,-46, 46, 90, 38,-54,-90,-31, 61, 88,22,-67,-85,-13, 73, 82, 4,-78,-78, 4, 82, 73,-13,-85,-67, 22, 88,61,-31,-90,-54, 38, 90, 46,-46, -90,-38, 54, 90, 31,-61,-88,-22, 67, 85,13,-73,-82, -4, 78,} {76,-11,-86,-62, 33, 90, 45,-52,-89,-24, 69, 83,2,-81,-71, 20, 88, 56,-41,-91,-37, 59, 87, 16, -74,-79, 7, 84,66,-28,-90,-48, 48, 90, 28,-66,-84, -7, 79, 74,-16,-87,-59, 37, 91,41,-56,-88,-20, 71, 81, -2,-83,-69, 24, 89, 52,-45,-90,-33, 62, 86,11,-76,} {75,-18,-89,-50, 50, 89, 18,-75,-75, 18, 89, 50,-50,-89,-18,75, 75,-18,-89,-50, 50, 89, 18, -75,-75, 18, 89, 50,-50,-89,-18, 75,75,-18,-89,-50, 50, 89, 18,-75,-75, 18, 89, 50,-50,-89,-18, 75,75,-18,-89,-50, 50, 89, 18,-75,-75, 18, 89, 50,-50,-89,-18, 75,}{74,-24,-90,-37, 66, 81,-11,-88,-48, 56, 86, 2,-84,-59, 45, 89,16,-79,-69, 33, 91, 28,-71,-76, 20, 90, 41,-62,-83, 7, 87, 52,-52,-87,-7, 83, 62,-41,-90,-20, 76, 71,-28,-91,-33, 69, 79,-16, -89,-45, 59, 84,-2,-86,-56, 48, 88, 11,-81,-66, 37, 90, 24,-74,} {73,-31,-90,-22, 78,67,-38,-90,-13, 82, 61,-46,-88, -4, 85, 54,-54,-85, 4, 88, 46,-61,-82,13, 90, 38,-67,-78, 22, 90, 31,-73,-73, 31, 90, 22,-78,-67, 38, 90,13,-82,-61, 46, 88, 4,-85,-54, 54, 85, -4,-88,-46, 61, 82,-13,-90,-38,67, 78,-22,-90,-31, 73,} {71,-37,-89, -7, 86, 48,-62,-79, 24, 91,20,-81,-59, 52, 84,-11,-90,-33, 74, 69,-41,-88, -2, 87, 45,-66,-76, 28,90, 16,-83,-56, 56, 83,-16,-90,-28, 76, 66,-45,-87, 2, 88, 41,-69,-74,33, 90, 11,-84,-52, 59, 81,-20,-91,-24, 79, 62,-48,-86, 7, 89, 37,-71,}{70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90, - 9, 87, 43,-70,-70, 43,87, -9,-90,-25, 80, 57, -57,-80, 25, 90, 9,-87,-43, 70, 70,-43,-87, 9,90, 25,-80,-57, 57, 80,-25,-90, -9, 87, 43,-70,-70, 43, 87, -9,-90,-25,80, 57,-57,-80, 25, 90, 9,-87,-43, 70,} {69,-48,-83, 24, 90, 2,-89,-28,81, 52,-66,-71, 45, 84,-20,-90, -7, 88, 33,-79,-56, 62, 74,-41, -86, 16,91, 11,-87,-37, 76, 59,-59,-76, 37, 87,-11,-91,-16, 86, 41,-74,-62, 56,79,-33,-88, 7, 90, 20,-84,-45, 71, 66,-52,-81, 28, 89, -2,-90,-24, 83,48,-69,} {67,-54,-78, 38, 85,-22,-90, 4, 90, 13,-88,-31, 82, 46,-73,-61,61, 73,-46,-82, 31, 88,-13,-90, -4, 90, 22,-85,-38, 78, 54,-67,-67, 54,78,-38,-85, 22, 90, -4,-90,-13, 88, 31,-82,-46, 73, 61, -61,-73, 46,82,-31,-88, 13, 90, 4,-90,-22, 85, 38,-78,-54, 67,} {66,-59,-71, 52,76,-45,-81, 37, 84,-28,-87, 20, 89,-11,-90, 2, 91, 7,-90,-16, 88,24,-86,-33, 83, 41,-79,-48, 74, 56,-69,-62, 62, 69,-56,-74, 48,79,-41,-83, 33, 86,-24,-88, 16, 90, -7,-91, -2, 90, 11,-89,-20, 87,28,-84,-37, 81, 45,-76,-52, 71, 59,-66,} {64,-64,-64, 64, 64,-64,-64,64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,64,-64,-64, 64, 64,-64,-64, 64,} {62,-69,-56, 74, 48,-79,-41, 83,33,-86,-24, 88, 16,-90,-7, 91,-2,-90, 11, 89,-20,-87, 28, 84, -37,-81,45, 76,-52,-71, 59, 66,-66,-59, 71, 52,-76,-45, 81, 37,-84,-28, 87,20,-89,-11, 90, 2, -91, 7, 90,-16,-88, 24, 86,-33,-83, 41, 79,-48,-74,56, 69,-62,} {61,-73,-46, 82, 31,-88,-13, 90, -4,-90, 22, 85,-38,-78,54, 67,-67,-54, 78, 38,-85,-22, 90, 4, -90, 13, 88,-31,-82, 46,73,-61,-61, 73, 46,-82,-31, 88, 13,-90, 4, 90,-22,-85, 38, 78,-54,-67,67, 54,-78,-38, 85, 22,-90, -4, 90,-13,-88, 31, 82,-46,-73, 61,}{59,-76,-37, 87, 11,-91, 16, 86,-41,-74, 62, 56,-79,-33, 88, 7,-90, 20,84,-45,-71, 66, 52,-81, -28, 89, 2,-90, 24, 83,-48,-69, 69, 48,-83,-24,90, -2,-89, 28, 81,-52,-66, 71, 45,-84,-20, 90, -7, -88, 33, 79,-56,-62,74, 41,-86,-16, 91,-11,-87, 37, 76,-59,} {57,-80,-25, 90, -9,-87, 43,70,-70,-43, 87, 9,-90, 25, 80,-57,-57, 80, 25,-90, 9, 87,-43,-70, 70,43,-87, -9, 90,-25,-80, 57, 57,-80,-25, 90, -9,-87, 43, 70,-70,-43, 87,9,-90, 25, 80,-57,-57, 80, 25,-90, 9, 87,-43,-70, 70, 43,-87, -9,90,-25,-80, 57,} {56,-83,-16, 90,-28,-76, 66, 45,-87, -2, 88,-41,-69,74, 33,-90, 11, 84,-52,-59, 81, 20,-91, 24, 79,-62,-48, 86, 7,-89, 37,71,-71,-37, 89, -7,-86, 48, 62,-79,-24, 91,-20,-81, 59, 52,-84,-11,90,-33,-74, 69, 41,-88, 2, 87,-45,-66, 76, 28,-90, 16, 83,-56,} {54,-85,-4, 88,-46,-61, 82, 13,-90, 38, 67,-78,-22, 90,-31,-73, 73, 31,-90, 22,78,-67,-38, 90, -13,-82, 61, 46,-88, 4, 85,-54,-54, 85, 4,-88, 46,61,-82,-13, 90,-38,-67, 78, 22,-90, 31, 73,-73, -31, 90,-22,-78, 67,38,-90, 13, 82,-61,-46, 88,-4,-85, 54,} {52,-87, 7, 83,-62,-41,90,-20,-76, 71, 28,-91, 33, 69,-79,-16, 89,-45,-59, 84, 2,-86, 56, 48,-88, 11, 81,-66,-37, 90,-24,-74, 74, 24,-90, 37, 66,-81,-11, 88,-48,-56,86, -2,-84, 59, 45,-89, 16, 79,-69,-33, 91,-28,-71, 76, 20,-90, 41,62,-83,-7, 87,-52,} {50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75,18,-89, 50, 50,-89, 18, 75,-75,-18, 89, -50,-50, 89,-18,-75, 75, 18,-89,50, 50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89, 50,50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89, 50,}{48,-90, 28, 66,-84, 7, 79,-74,-16, 87,-59,-37, 91,-41,-56, 88,-20,-71,81, 2,-83, 69, 24,-89, 52, 45,-90, 33, 62,-86, 11, 76,-76,-11,86,-62,-33, 90,-45,-52, 89,-24,-69, 83, -2,-81, 71, 20, -88, 56, 41,-91,37, 59,-87, 16, 74,-79, -7, 84,-66,-28, 90,-48,} {46,-90, 38, 54,-90,31, 61,-88, 22, 67,-85, 13, 73,-82, 4, 78,-78, -4, 82,-73,-13,85,-67,-22, 88,-61,-31, 90,-54,-38, 90,-46,-46, 90,-38,-54, 90,-31,-61,88,-22,-67, 85,-13,-73, 82, -4,-78, 78, 4,-82, 73, 13,-85, 67, 22,-88,61, 31,-90, 54, 38,-90, 46,} {45,-91, 48, 41,-90, 52, 37,-90, 56,33,-89, 59, 28,-88, 62, 24,-87, 66, 20,-86, 69, 16,-84, 71, 11,-83, 74,7,-81, 76, 2,-79, 79, -2,-76, 81,-7,-74, 83,-11,-71, 84,-16,-69,86,-20,-66, 87,-24, -62, 88,-28,-59, 89,-33,-56, 90,-37,-52, 90,-41,-48,91,-45,} {43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70, 87,-25,-57,90,-43,-43, 90,-57,-25, 87,-70, -9, 80, -80, 9, 70,-87, 25, 57,-90, 43,43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70, 87,-25,-57, 90,-43,-43,90,-57,-25, 87,-70, -9, 80,-80, 9, 70,-87, 25, 57,-90, 43,} {41,-89, 66,11,-79, 83,-20,-59, 90,-48,-33, 87,-71, -2, 74,-86, 28, 52,-91, 56,24,-84, 76, -7, -69, 88,-37,-45, 90,-62,-16, 81,-81, 16, 62,-90, 45,37,-88, 69, 7,-76, 84,-24,-56, 91,-52,-28, 86,-74, 2, 71,-87, 33,48,-90, 59, 20,-83, 79,-11,-66, 89,-41,} {38,-88, 73,-4,-67, 90,-46,-31,85,-78, 13, 61,-90, 54, 22,-82, 82,-22,-54, 90,-61,-13, 78,-85, 31,46,-90, 67, 4,-73, 88,-38,-38, 88,-73, 4, 67,-90, 46, 31,-85,78,-13,-61, 90,-54,-22, 82,-82, 22, 54,-90, 61, 13,-78, 85,-31,-46,90,-67, -4, 73,-88, 38,} {37,-86, 79,-20,-52, 90,-69, 2, 66,-90, 56,16,-76, 87,-41,-33, 84,-81, 24, 48,-89, 71,-7,-62, 91,-59,-11, 74,-88,45, 28,-83, 83,-28,-45, 88,-74, 11, 59,-91, 62, 7,-71, 89,-48,-24,81,-84, 33, 41,-87, 76,-16,-56, 90,-66, -2, 69,-90, 52, 20,-79, 86,-37,}{36,-83, 83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {33,-81,87,-48,-16, 71,-90, 62, -2,-59, 90,-74, 20, 45,-86, 83,-37,-28, 79,-88,52, 11,-69, 91, -66, 7, 56,-89, 76,-24,-41, 84,-84, 41, 24,-76, 89,-56,-7, 66,-91, 69,-11,-52, 88,-79, 28, 37,-83, 86,-45,-20, 74,-90, 59,2,-62, 90,-71, 16, 48,-87, 81,-33,} {31,-78, 90,-61, 4, 54,-88,82,-38,-22, 73,-90, 67,-13,-46, 85,-85, 46, 13,-67, 90,-73, 22, 38, -82,88,-54, -4, 61,-90, 78,-31,-31, 78,-90, 61, -4,-54, 88,-82, 38, 22,-73,90,-67, 13, 46,-85, 85,-46,-13, 67,-90, 73,-22,-38, 82,-88, 54, 4,-61,90,-78, 31,} {28,-74, 91,-71, 24, 33,-76, 90,-69, 20, 37,-79, 90,-66,16, 41,-81, 89,-62, 11, 45,-83, 88,-59, 7, 48,-84, 87,-56, 2, 52,-86,86,-52, -2, 56,-87, 84,-48, -7, 59,-88, 83,-45,-11, 62,-89, 81,-41, -16,66,-90, 79,-37,-20, 69,-90, 76,-33,-24, 71,-91, 74,-28,} {25,-70,90,-80, 43, 9,-57, 87,-87, 57,-9,-43, 80,-90, 70,-25,-25, 70,-90,80,-43, -9, 57,-87, 87,-57, 9, 43,-80, 90,-70, 25, 25,-70, 90,-80, 43,9,-57, 87,-87, 57, -9,-43, 80,-90, 70,-25,-25, 70,-90, 80,-43, -9,57,-87, 87,-57, 9, 43,-80, 90,-70, 25,} {24,-66, 88,-86, 59,-16,-33,71,-90, 83,-52, 7, 41,-76, 91,-79, 45, 2,-48, 81,-90, 74,-37,-11,56,-84, 89,-69, 28, 20,-62, 87,-87, 62,-20,-28, 69,-89, 84,-56, 11,37,-74, 90,-81, 48, -2,-45, 79,-91, 76,-41, -7, 52,-83, 90,-71, 33,16,-59, 86,-88, 66,-24,} {22,-61, 85,-90, 73,-38, -4, 46,-78, 90,-82,54,-13,-31, 67,-88, 88,-67, 31, 13,-54, 82,-90, 78, -46, 4, 38,-73,90,-85, 61,-22,-22, 61,-85, 90,-73, 38, 4,-46, 78,-90, 82,-54, 13,31,-67, 88,-88, 67,-31,-13, 54,-82, 90,-78, 46, -4,-38, 73,-90, 85,-61,22,} {20,-56, 81,-91, 83,-59, 24, 16,-52, 79,-90, 84,-62, 28, 11,-48,76,-90, 86,-66, 33, 7,-45, 74, -89, 87,-69, 37, 2,-41, 71,-88, 88,-71,41,-2,-37, 69,-87, 89,-74, 45, -7,-33, 66,-86, 90,-76, 48, -11,-28,62,-84, 90,-79, 52,-16,-24, 59,-83, 91,-81, 56,-20,} {18,-50, 75,-89,89,-75, 50,-18,-18, 50,-75, 89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75,50, -18,-18, 50,-75, 89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75,50,-18,-18, 50,-75, 89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75,50,-18,-18, 50,-75, 89,-89, 75,-50, 18,} {16,-45, 69,-84, 91,-86,71,-48, 20, 11,-41, 66,-83, 90,-87, 74,-52, 24, 7,-37, 62,-81, 90,-88,76,-56, 28, 2,-33, 59,-79, 89,-89, 79,-59, 33,-2,-28, 56,-76, 88,-90,81,-62, 37,-7,-24, 52,-74, 87,-90, 83,-66, 41,-11,-20, 48,-71, 86,-91,84,-69, 45,-16,} {13,-38, 61,-78, 88,-90, 85,-73, 54,-31, 4, 22,-46,67,-82, 90,-90, 82,-67, 46,-22, -4, 31,-54, 73,-85, 90,-88, 78,-61,38,-13,-13, 38,-61, 78,-88, 90,-85, 73,-54, 31,-4,-22, 46,-67, 82,-90,90,-82, 67,-46, 22, 4,-31, 54,-73, 85,-90, 88,-78, 61,-38, 13,} {11,-33,52,-69, 81,-88, 91,-87, 79,-66, 48,-28, 7, 16,-37, 56,-71, 83,-89,90,-86, 76,-62, 45, -24, 2, 20,-41, 59,-74, 84,-90, 90,-84, 74,-59,41,-20, -2, 24,-45, 62,-76, 86,-90, 89,-83, 71,-56, 37,-16, -7, 28,-48,66,-79, 87,-91, 88,-81, 69,-52, 33,-11,} {9,-25, 43,-57, 70,-80, 87,-90,90,-87, 80,-70, 57,-43, 25,-9, -9, 25,-43, 57,-70, 80,-87, 90, -90,87,-80, 70,-57, 43,-25, 9, 9,-25, 43,-57, 70,-80, 87,-90, 90,-87,80,-70, 57,-43, 25, -9, -9, 25,-43, 57,-70, 80,-87, 90,-90, 87,-80,70,-57, 43,-25, 9,} {7,-20, 33,-45, 56,-66, 74,-81, 86,-89, 91,-90,87,-83, 76,-69, 59,-48, 37,-24, 11, 2,-16, 28, -41, 52,-62, 71,-79,84,-88, 90,-90, 88,-84, 79,-71, 62,-52, 41,-28, 16, -2,-11, 24,-37,48,-59, 69,-76, 83,-87, 90,-91, 89,-86, 81,-74, 66,-56, 45,-33, 20,-7,}{4,-13, 22,-31, 38,-46, 54,-61, 67,-73, 78,-82, 85,-88, 90,-90, 90,-90,88,-85, 82,-78, 73,-67, 61,-54, 46,-38, 31,-22, 13,-4, -4, 13,-22,31,-38, 46,-54, 61,-67, 73,-78, 82,-85, 88,-90, 90, -90, 90,-88, 85,-82,78,-73, 67,-61, 54,-46, 38,-31, 22,-13, 4,} {2, -7, 11,-16, 20,-24,28,-33, 37,-41, 45,-48, 52,-56, 59,-62, 66,-69, 71,-74, 76,-79, 81,-83,84,-86, 87,-88, 89,-90, 90,-91, 91,-90, 90,-89, 88,-87, 86,-84, 83,-81,79,-76, 74,-71, 69,-66, 62,-59, 56,-52, 48,-45, 41,-37, 33,-28, 24,-20,16,-11, 7,-2,}.
 15. A non-transitory computer readable storage devicehaving computer readable instructions stored therein that when executedby a processor perform a video decoder method comprising: inversetransforming with a processing device a first block of transformcoefficients by a transform matrix having a plurality of basis vectorsinto a corresponding second block of residual pixel values for a videoimage, wherein the transform matrix consists of integer elements, hasbasis vectors derived from one of a DCT and a KLT, has a DC vector, notall of the basis vectors being orthogonal but all of the basis vectorsbeing within a predetermined deviation of the DC vector, norms of thebasis vectors having respective deviations with respect to said DCvector less than 0.0015, and at least one norm-deviation being non-zero,having a maximum element deviation from said one of a DCT and a KLT lessthan or equal to 0.05, and at least one element deviation beingnon-zero, and each integer element being represented in one byte. 16.The computer readable storage device of claim 15, wherein any innerproduct between two of the basis vectors is less than 0.003.
 17. Thecomputer readable storage device of claim 15, wherein said transformmatrix is a 4×4 transform matrix and having basis vectors defined as:{64, 64, 64, 64,} {83, 36,-36,-83,} {64,-64, -64, 64,} {36,-83,83,-36,}.
 18. The computer readable storage device of claim 15, whereinsaid transform matrix is a 8×8 transform matrix and having basis vectorsdefined as: {64, 64, 64, 64, 64, 64, 64, 64,} {89, 75, 50,18,-18,-50,-75,-89,} {83, 36,-36,-83,-83,-36, 36, 83,} {75,-18,-89,-50,50, 89, 18,-75,} {64,-64,-64, 64, 64,-64,-64, 64,} {50,-89, 18,75,-75,-18, 89,-50,} {36,-83, 83,-36,-36, 83,-83, 36,} {18,-50, 75,-89,89,-75, 50,-18,}.
 19. The computer readable storage device of claim 15,wherein said transform matrix is a 16×16 transform matrix and havingbasis vectors defined as: {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64,} {90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89,} {87, 57,9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43, -9,-57,-87,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {80,9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70, -9,-80,}{75,-18,-89,-50, 50, 89, 18,-75,-75, 18, 89, 50,-50,-89,-18, 75,}{70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90,-9, 87, 43,-70,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{57,-80,-25, 90,-9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,} {50,-89,18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89, 50,} {43,-90, 57,25,-87, 70, 9,-80, 80, -9,-70, 87,-25,-57, 90,-43,} {36,-83, 83,-36,-36,83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {25,-70, 90,-80, 43, 9,-57,87,-87, 57,-9,-43, 80,-90, 70,-25,} {18,-50, 75,-89, 89,-75, 50,-18,-18,50,-75, 89,-89, 75,-50, 18,} {9,-25, 43,-57, 70,-80, 87,-90, 90,-87,80,-70, 57,-43, 25,-9,}.
 20. The computer readable storage device ofclaim 15, wherein said transform matrix is a 32×32 transform matrix andhaving basis vectors defined as: {64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64,} {90, 90, 88, 85, 82, 78, 73, 67, 61, 54, 46, 38,31, 22, 13, 4, -4,-13,-22,-31,-38,-46,-54,-61,-67,-73,-78,-82,-85,-88,-90,-90,} {90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90,-87,-80,-70,-57,-43,-25, -9, 9, 25,43, 57, 70, 80, 87, 90,} {90, 82, 67, 46, 22,-4,-31,-54,-73,-85,-90,-88,-78,-61,-38,-13, 13, 38, 61, 78, 88, 90, 85,73, 54, 31, 4,-22,-46,-67,-82,-90,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75, -89,-89,-75,-50,-18, 18, 50, 75, 89,} {88, 67,31,-13,-54,-82,-90,-78,-46, -4, 38, 73, 90, 85, 61,22,-22,-61,-85,-90,-73,-38, 4, 46, 78, 90, 82, 54, 13,-31,-67,-88,} {87,57, 9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43,-9,-57,-87,-87,-57,-9, 43,80, 90, 70, 25, -25,-70,-90,-80,-43, 9, 57, 87,} {85,46,-13,-67,-90,-73,-22, 38, 82, 88, 54, -4,-61,-90,-78,-31, 31, 78, 90,61, 4,-54,-88,-82, -38, 22, 73, 90, 67, 13,-46,-85,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {82,22,-54,-90,-61, 13, 78, 85, 31,-46,-90,-67, 4, 73, 88, 38,-38,-88,-73,-4, 67, 90, 46,-31, -85,-78,-13, 61, 90, 54,-22,-82,} {80,9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70, -9,-80,-80, -9, 70,87, 25,-57,-90,-43, 43, 90, 57,-25,-87,-70, 9, 80,} {78,-4,-82,-73, 13,85, 67,-22,-88,-61, 31, 90, 54,-38,-90,-46, 46, 90, 38,-54,-90,-31, 61,88, 22,-67,-85,-13, 73, 82, 4,-78,} {75,-18,-89,-50, 50, 89, 18,-75,-75,18, 89, 50,-50,-89,-18, 75, 75,-18,-89,-50, 50, 89, 18, -75,-75, 18, 89,50,-50,-89,-18, 75,} {73,-31,-90,-22, 78, 67,-38,-90,-13, 82,61,-46,-88, -4, 85, 54,-54,-85, 4, 88, 46,-61,-82, 13, 90, 38,-67,-78,22, 90, 31,-73,} {70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90, -9, 87,43,-70,-70, 43, 87, -9,-90,-25, 80, 57, -57,-80, 25, 90, 9,-87,-43, 70,}{67,-54,-78, 38, 85,-22,-90, 4, 90, 13,-88,-31, 82, 46,-73,-61, 61,73,-46,-82, 31, 88,-13,-90, -4, 90, 22,-85,-38, 78, 54,-67,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{61,-73,-46, 82, 31,-88,-13, 90, -4,-90, 22, 85,-38,-78, 54, 67,-67,-54,78, 38,-85,-22, 90, 4, -90, 13, 88,-31,-82, 46, 73,-61,} {57,-80,-25,90, -9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,-57, 80, 25,-90, 9,87,-43,-70, 70, 43,-87, -9, 90,-25,-80, 57,} {54,-85,-4, 88,-46,-61, 82,13,-90, 38, 67,-78,-22, 90,-31,-73, 73, 31,-90, 22, 78,-67,-38, 90,-13,-82, 61, 46,-88, 4, 85,-54,} {50,-89, 18, 75,-75,-18, 89,-50,-50,89,-18,-75, 75, 18,-89, 50, 50,-89, 18, 75,-75,-18, 89, -50,-50,89,-18,-75, 75, 18,-89, 50,} {46,-90, 38, 54,-90, 31, 61,-88, 22,67,-85, 13, 73,-82, 4, 78,-78, -4, 82,-73,-13, 85,-67,-22, 88,-61,-31,90,-54,-38, 90,-46,} {43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70,87,-25,-57, 90,-43,-43, 90,-57,-25, 87,-70, -9, 80, -80, 9, 70,-87, 25,57,-90, 43,} {38,-88, 73,-4,-67, 90,-46,-31, 85,-78, 13, 61,-90, 54,22,-82, 82,-22,-54, 90,-61,-13, 78,-85, 31, 46,-90, 67, 4,-73, 88,-38,}{36,-83, 83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {31,-78,90,-61, 4, 54,-88, 82,-38,-22, 73,-90, 67,-13,-46, 85,-85, 46, 13,-67,90,-73, 22, 38, -82, 88,-54, -4, 61,-90, 78,-31,} {25,-70, 90,-80, 43,9,-57, 87,-87, 57,-9,-43, 80,-90, 70,-25,-25, 70,-90, 80,-43, -9,57,-87, 87,-57, 9, 43,-80, 90,-70, 25,} {22,-61, 85,-90, 73,-38, -4,46,-78, 90,-82, 54,-13,-31, 67,-88, 88,-67, 31, 13,-54, 82,-90, 78, -46,4, 38,-73, 90,-85, 61,-22,} {18,-50, 75,-89, 89,-75, 50,-18,-18, 50,-75,89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75, 50, -18,-18, 50,-75, 89,-89,75,-50, 18,} {13,-38, 61,-78, 88,-90, 85,-73, 54,-31, 4, 22,-46, 67,-82,90,-90, 82,-67, 46,-22, -4, 31,-54, 73,-85, 90,-88, 78,-61, 38,-13,}{9,-25, 43,-57, 70,-80, 87,-90, 90,-87, 80,-70, 57,-43, 25, -9, -9,25,-43, 57,-70, 80,-87, 90, -90, 87,-80, 70,-57, 43,-25, 9,} {4,-13,22,-31, 38,-46, 54,-61, 67,-73, 78,-82, 85,-88, 90,-90, 90,-90, 88,-85,82,-78, 73,-67, 61,-54, 46,-38, 31,-22, 13, -4,}.
 21. The computerreadable storage device of claim 15, wherein said transform matrix is a64×64 transform matrix and having basis vectors defined as: {64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64,} {91, 90, 90, 89, 88, 87, 86, 84, 83, 81,79, 76, 74, 71, 69, 66, 62, 59, 56, 52, 48, 45, 41, 37, 33, 28, 24, 20,16, 11, 7, 2, -2,-7,-11,-16,-20,-24,-28,-33,-37,-41,-45,-48,-52,-56,-59,-62,-66,-69,-71,-74,-76,-79,-81,-83,-84,-86,-87,-88,-89,-90,-90,-91,} {90, 90,88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4,-4,-13,-22,-31,-38,-46,-54,-61,-67,-73,-78,-82,-85,-88,-90,-90,-90,-90,-88,-85,-82,-78,-73,-67,-61,-54,-46,-38,-31,-22,-13,-4, 4, 13, 22, 31, 38, 46, 54, 61, 67, 73, 78, 82, 85, 88, 90, 90,} {90,88, 84, 79, 71, 62, 52, 41, 28, 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88,-54,-4, 61,-90, 78,-31,-31, 78,-90, 61,-4,-54, 88,-82, 38, 22,-73, 90,-67, 13, 46,-85, 85,-46,-13, 67,-90,73,-22,-38, 82,-88, 54, 4,-61, 90,-78, 31,} {28,-74, 91,-71, 24, 33,-76,90,-69, 20, 37,-79, 90,-66, 16, 41,-81, 89,-62, 11, 45,-83, 88,-59, 7,48,-84, 87,-56, 2, 52,-86, 86,-52, -2, 56,-87, 84,-48, -7, 59,-88,83,-45,-11, 62,-89, 81,-41, -16, 66,-90, 79,-37,-20, 69,-90, 76,-33,-24,71,-91, 74,-28,} {25,-70, 90,-80, 43, 9,-57, 87,-87, 57,-9,-43, 80,-90,70,-25,-25, 70,-90, 80,-43, -9, 57,-87, 87,-57, 9, 43,-80, 90,-70, 25,25,-70, 90,-80, 43, 9,-57, 87,-87, 57,-9,-43, 80,-90, 70,-25,-25,70,-90, 80,-43, -9, 57,-87, 87,-57, 9, 43,-80, 90,-70, 25,} {24,-66,88,-86, 59,-16,-33, 71,-90, 83,-52, 7, 41,-76, 91,-79, 45, 2,-48,81,-90, 74,-37,-11, 56,-84, 89,-69, 28, 20,-62, 87,-87, 62,-20,-28,69,-89, 84,-56, 11, 37,-74, 90,-81, 48, -2,-45, 79,-91, 76,-41, -7,52,-83, 90,-71, 33, 16,-59, 86,-88, 66,-24,} {22,-61, 85,-90, 73,-38,-4, 46,-78, 90,-82, 54,-13,-31, 67,-88, 88,-67, 31, 13,-54, 82,-90, 78,-46, 4, 38,-73, 90,-85, 61,-22,-22, 61,-85, 90,-73, 38, 4,-46, 78,-90,82,-54, 13, 31,-67, 88,-88, 67,-31,-13, 54,-82, 90,-78, 46, -4,-38,73,-90, 85,-61, 22,} {20,-56, 81,-91, 83,-59, 24, 16,-52, 79,-90,84,-62, 28, 11,-48, 76,-90, 86,-66, 33, 7,-45, 74, -89, 87,-69, 37,2,-41, 71,-88, 88,-71, 41, -2,-37, 69,-87, 89,-74, 45, -7,-33, 66,-86,90,-76, 48, -11,-28, 62,-84, 90,-79, 52,-16,-24, 59,-83, 91,-81,56,-20,} {18,-50, 75,-89, 89,-75, 50,-18,-18, 50,-75, 89,-89, 75,-50,18, 18,-50, 75,-89, 89,-75, 50, -18,-18, 50,-75, 89,-89, 75,-50, 18,18,-50, 75,-89, 89,-75, 50,-18,-18, 50,-75, 89,-89, 75,-50, 18, 18,-50,75,-89, 89,-75, 50,-18,-18, 50,-75, 89,-89, 75,-50, 18,} {16,-45,69,-84, 91,-86, 71,-48, 20, 11,-41, 66,-83, 90,-87, 74,-52, 24, 7,-37,62,-81, 90,-88, 76,-56, 28, 2,-33, 59,-79, 89,-89, 79,-59, 33,-2,-28,56,-76, 88,-90, 81,-62, 37,-7,-24, 52,-74, 87,-90, 83,-66, 41,-11,-20,48,-71, 86,-91, 84,-69, 45,-16,} {13,-38, 61,-78, 88,-90, 85,-73,54,-31, 4, 22,-46, 67,-82, 90,-90, 82,-67, 46,-22, -4, 31,-54, 73,-85,90,-88, 78,-61, 38,-13,-13, 38,-61, 78,-88, 90,-85, 73,-54, 31, -4,-22,46,-67, 82,-90, 90,-82, 67,-46, 22, 4,-31, 54,-73, 85,-90, 88,-78,61,-38, 13,} {11,-33, 52,-69, 81,-88, 91,-87, 79,-66, 48,-28, 7, 16,-37.56,-71, 83,-89, 90,-86, 76,-62, 45, -24, 2, 20,-41, 59,-74, 84,-90,90,-84, 74,-59, 41,-20, -2, 24,-45, 62,-76, 86,-90, 89,-83, 71,-56,37,-16,-7, 28,-48, 66,-79, 87,-91, 88,-81, 69,-52, 33,-11,} {9,-25,43,-57, 70,-80, 87,-90, 90,-87, 80,-70, 57,-43, 25, -9, -9, 25,-43,57,-70, 80,-87, 90, -90, 87,-80, 70,-57, 43,-25, 9, 9,-25, 43,-57,70,-80, 87,-90, 90,-87, 80,-70, 57,-43, 25, -9, -9, 25,-43, 57,-70,80,-87, 90,-90, 87,-80, 70,-57, 43,-25, 9,} {7,-20, 33,-45, 56,-66,74,-81, 86,-89, 91,-90, 87,-83, 76,-69, 59,-48, 37,-24, 11, 2,-16, 28,-41, 52,-62, 71,-79, 84,-88, 90,-90, 88,-84, 79,-71, 62,-52, 41,-28, 16,-2,-11, 24,-37, 48,-59, 69,-76, 83,-87, 90,-91, 89,-86, 81,-74, 66,-56,45,-33, 20, -7,} {4,-13, 22,-31, 38,-46, 54,-61, 67,-73, 78,-82, 85,-88,90,-90, 90,-90, 88,-85, 82,-78, 73,-67, 61,-54, 46,-38, 31,-22, 13, -4,-4, 13,-22, 31,-38, 46,-54, 61,-67, 73,-78, 82,-85, 88,-90, 90, -90,90,-88, 85,-82, 78,-73, 67,-61, 54,-46, 38,-31, 22,-13, 4,} {2, -7,11,-16, 20,-24, 28,-33, 37,-41, 45,-48, 52,-56, 59,-62, 66,-69, 71,-74,76,-79, 81,-83, 84,-86, 87,-88, 89,-90, 90,-91, 91,-90, 90,-89, 88,-87,86,-84, 83,-81, 79,-76, 74,-71, 69,-66, 62,-59, 56,-52, 48,-45, 41,-37,33,-28, 24,-20, 16,-11, 7, -2,}.
 22. A method of video codingcomprising: transforming with a processing device a first block ofresidual pixel values of an image by a transform matrix having aplurality of basis vectors into a corresponding second block oftransform coefficients, wherein the transform matrix consists of integerelements, has basis vectors derived from one of a DCT and a KLT, has aDC vector, not all of the basis vectors being orthogonal but all of thebasis vectors being within a predetermined deviation of the DC vector,norms of the basis vectors having respective deviations with respect tosaid DC vector less than 0.0015, and at least one norm-deviation beingnon-zero, having a maximum element deviation from said one of a DCT anda KLT less than or equal to 0.05, and at least one element deviationbeing non-zero, and each integer element being represented in one byte.23. The method of claim 22, wherein any inner product between two of thebasis vectors is less than 0.003.
 24. The method according to claim 22,wherein said transform matrix is a 4×4 transform matrix and having basisvectors defined as: {64, 64, 64, 64,} {83, 36,-36,-83,} {64,-64,-64,64,} {36,-83, 83,-36,}.
 25. The method according to claim 22, whereinsaid transform matrix is a 8×8 transform matrix and having basis vectorsdefined as: {64, 64, 64, 64, 64, 64, 64, 64,} {89, 75, 50,18,-18,-50,-75,-89,} {83, 36,-36,-83,-83,-36, 36, 83,} {75,-18,-89,-50,50, 89, 18,-75,} {64,-64,-64, 64, 64,-64,-64, 64,} {50,-89, 18,75,-75,-18, 89,-50,} {36,-83, 83,-36,-36, 83,-83, 36,} {18,-50, 75,-89,89,-75, 50,-18,}.
 26. The method according to claim 22, wherein saidtransform matrix is a 16×16 transform matrix and having basis vectorsdefined as: {64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64,} {90, 87, 80, 70, 57, 43, 25, 9, -9,-25,-43,-57,-70,-80,-87,-90,}{89, 75, 50, 18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89,} {87,57, 9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43,-9,-57,-87,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {80,9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70,-9,-80,}{75,-18,-89,-50, 50, 89, 18,-75,-75, 18, 89, 50,-50,-89,-18, 75,}{70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90,-9, 87, 43,-70,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{57,-80,-25, 90, -9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,}{50,-89, 18, 75,-75,-18, 89,-50,-50, 89,-18,-75, 75, 18,-89, 50,}{43,-90, 57, 25,-87, 70, 9,-80, 80,-9,-70, 87,-25,-57, 90,-43,} {36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {25,-70,90,-80, 43, 9,-57, 87,-87, 57,-9,-43, 80,-90, 70,-25,} {18,-50, 75,-89,89,-75, 50,-18,-18, 50,-75, 89,-89, 75,-50, 18,} {9,-25, 43,-57, 70,-80,87,-90, 90,-87, 80,-70, 57,-43, 25, -9,}.
 27. The method according toclaim 22, wherein said transform matrix is a 32×32 transform matrix andhaving basis vectors defined as: {64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64,} {90, 90, 88, 85, 82, 78, 73, 67, 61, 54, 46, 38,31, 22, 13, 4, -4,-13,-22,-31,-38,-46,-54,-61,-67,-73,-78,-82,-85,-88,-90,-90,} {90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90,-87,-80,-70,-57,-43,-25,-9, 9, 25,43, 57, 70, 80, 87, 90,} {90, 82, 67, 46, 22,-4,-31,-54,-73,-85,-90,-88,-78,-61,-38,-13, 13, 38, 61, 78, 88, 90, 85,73, 54, 31, 4,-22,-46,-67,-82,-90,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75, -89,-89,-75,-50,-18, 18, 50, 75, 89,} {88, 67,31,-13,-54,-82,-90,-78,-46, -4, 38, 73, 90, 85, 61,22,-22,-61,-85,-90,-73,-38, 4, 46, 78, 90, 82, 54, 13,-31,-67,-88,} {87,57, 9,-43,-80,-90,-70,-25, 25, 70, 90, 80, 43, -9,-57,-87,-87,-57, -9,43, 80, 90, 70, 25, -25,-70,-90,-80,-43, 9, 57, 87,} {85,46,-13,-67,-90,-73,-22, 38, 82, 88, 54, -4,-61,-90,-78,-31, 31, 78, 90,61, 4,-54,-88,-82, -38, 22, 73, 90, 67, 13,-46,-85,} {83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83, 83,36,-36,-83,-83,-36, 36, 83, 83, 36,-36,-83,-83,-36, 36, 83,} {82,22,-54,-90,-61, 13, 78, 85, 31,-46,-90,-67, 4, 73, 88, 38,-38,-88,-73,-4, 67, 90, 46,-31, -85,-78,-13, 61, 90, 54,-22,-82,} {80,9,-70,-87,-25, 57, 90, 43,-43,-90,-57, 25, 87, 70, -9,-80,-80, -9, 70,87, 25,-57,-90,-43, 43, 90, 57,-25,-87,-70, 9, 80,} {78, -4,-82,-73, 13,85, 67,-22,-88,-61, 31, 90, 54,-38,-90,-46, 46, 90, 38,-54,-90,-31, 61,88, 22,-67,-85,-13, 73, 82, 4,-78,} {75,-18,-89,-50, 50, 89, 18,-75,-75,18, 89, 50,-50,-89,-18, 75, 75,-18,-89,-50, 50, 89, 18, -75,-75, 18, 89,50,-50,-89,-18, 75,} {73,-31,-90,-22, 78, 67,-38,-90,-13, 82,61,-46,-88, -4, 85, 54,-54,-85, 4, 88, 46,-61,-82, 13, 90, 38,-67,-78,22, 90, 31,-73,} {70,-43,-87, 9, 90, 25,-80,-57, 57, 80,-25,-90, -9, 87,43,-70,-70, 43, 87, -9,-90,-25, 80, 57, -57,-80, 25, 90, 9,-87,-43, 70,}{67,-54,-78, 38, 85,-22,-90, 4, 90, 13,-88,-31, 82, 46,-73,-61, 61,73,-46,-82, 31, 88,-13,-90, -4, 90, 22,-85,-38, 78, 54,-67,}{64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64, 64,-64,-64, 64,}{61,-73,-46, 82, 31,-88,-13, 90, -4,-90, 22, 85,-38,-78, 54, 67,-67,-54,78, 38,-85,-22, 90, 4, -90, 13, 88,-31,-82, 46, 73,-61,} {57,-80,-25,90,-9,-87, 43, 70,-70,-43, 87, 9,-90, 25, 80,-57,-57, 80, 25,-90, 9,87,-43,-70, 70, 43,-87, -9, 90,-25,-80, 57,} {54,-85, -4, 88,-46,-61,82, 13,-90, 38, 67,-78,-22, 90,-31,-73, 73, 31,-90, 22, 78,-67,-38, 90,-13,-82, 61, 46,-88, 4, 85,-54,} {50,-89, 18, 75,-75,-18, 89,-50,-50,89,-18,-75, 75, 18,-89, 50, 50,-89, 18, 75,-75,-18, 89, -50,-50,89,-18,-75, 75, 18,-89, 50,} {46,-90, 38, 54,-90, 31, 61,-88, 22,67,-85, 13, 73,-82, 4, 78,-78, -4, 82,-73,-13, 85,-67,-22, 88,-61,-31,90,-54,-38, 90,-46,} {43,-90, 57, 25,-87, 70, 9,-80, 80, -9,-70,87,-25,-57, 90,-43,-43, 90,-57,-25, 87,-70, -9, 80, -80, 9, 70,-87, 25,57,-90, 43,} {38,-88, 73, -4,-67, 90,-46,-31, 85,-78, 13, 61,-90, 54,22,-82, 82,-22,-54, 90,-61,-13, 78,-85, 31, 46,-90, 67, 4,-73, 88,-38,}{36,-83, 83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36, 36,-83,83,-36,-36, 83,-83, 36, 36,-83, 83,-36,-36, 83,-83, 36,} {31,-78,90,-61, 4, 54,-88, 82,-38,-22, 73,-90, 67,-13,-46, 85,-85, 46, 13,-67,90,-73, 22, 38, -82, 88,-54, -4, 61,-90, 78,-31,} {25,-70, 90,-80, 43,9,-57, 87,-87, 57,-9,-43, 80,-90, 70,-25,-25, 70,-90, 80,-43, -9,57,-87, 87,-57, 9, 43,-80, 90,-70, 25,} {22,-61, 85,-90, 73,-38, -4,46,-78, 90,-82, 54,-13,-31, 67,-88, 88,-67, 31, 13,-54, 82,-90, 78, -46,4, 38,-73, 90,-85, 61,-22,} {18,-50, 75,-89, 89,-75, 50,-18,-18, 50,-75,89,-89, 75,-50, 18, 18,-50, 75,-89, 89,-75, 50, -18,-18, 50,-75, 89,-89,75,-50, 18,} {13,-38, 61,-78, 88,-90, 85,-73, 54,-31, 4, 22,-46, 67,-82,90,-90, 82,-67, 46,-22, -4, 31,-54, 73,-85, 90,-88, 78,-61, 38,-13,}{9,-25, 43,-57, 70,-80, 87,-90, 90,-87, 80,-70, 57,-43, 25,-9,-9,25,-43, 57,-70, 80,-87, 90, -90, 87,-80, 70,-57, 43,-25, 9,} {4,-13,22,-31, 38,-46, 54,-61, 67,-73, 78,-82, 85,-88, 90,-90, 90,-90, 88,-85,82,-78, 73,-67, 61,-54, 46,-38, 31,-22, 13, -4,}.
 28. The methodaccording to claim 22, wherein said transform matrix is a 64×64transform matrix and having basis vectors defined as: {64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64,} {91, 90, 90, 89, 88, 87, 86, 84, 83, 81, 79,76, 74, 71, 69, 66, 62, 59, 56, 52, 48, 45, 41, 37, 33, 28, 24, 20, 16,11, 7, 2, -2,-7,-11,-16,-20,-24,-28,-33,-37,-41,-45,-48,-52,-56,-59,-62,-66,-69,-71,-74,-76,-79,-81,-83,-84,-86,-87,-88,-89,-90,-90,-91,} {90, 90,88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4,-4,-13,-22,-31,-38,-46,-54,-61,-67,-73,-78,-82,-85,-88,-90,-90,-90,-90,-88,-85,-82,-78,-73,-67,-61,-54,-46,-38,-31,-22,-13,-4, 4, 13, 22, 31, 38, 46, 54, 61, 67, 73, 78, 82, 85, 88, 90, 90,} {90,88, 84, 79, 71, 62, 52, 41, 28, 16,2,-11,-24,-37,-48,-59,-69,-76,-83,-87,-90,-91,-89,-86,-81,-74,-66,-56,-45,-33,-20, -7, 7, 20, 33, 45, 56, 66, 74, 81, 86, 89,91, 90, 87, 83, 76, 69, 59, 48, 37, 24, 11,-2,-16,-28,-41,-52,-62,-71,-79,-84,-88,-90,} {90, 87, 80, 70, 57, 43,25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90,-87,-80,-70,-57,-43,-25, -9, 9,25, 43, 57, 70, 80, 87, 90, 90, 87, 80, 70, 57, 43, 25, 9,-9,-25,-43,-57,-70,-80,-87,-90,-90, -87,-80,-70,-57,-43,-25, -9, 9, 25,43, 57, 70, 80, 87, 90,} {90, 84, 74, 59, 41, 20,-2,-24,-45,-62,-76,-86,-90,-89,-83,-71,-56,-37,-16, 7, 28, 48, 66, 79,87, 91, 88, 81, 69, 52, 33,11,-11,-33,-52,-69,-81,-88,-91,-87,-79,-66,-48,-28, -7, 16, 37, 56, 71,83, 89, 90, 86, 76, 62, 45, 24, 2,-20,-41,-59,-74,-84,-90,} {90, 82, 67,46, 22, -4,-31,-54,-73,-85,-90,-88,-78,-61,-38,-13, 13, 38, 61, 78, 88,90, 85, 73, 54, 31, 4,-22,-46,-67,-82,-90,-90,-82,-67,-46,-22, 4, 31,54, 73, 85, 90, 88, 78, 61, 38, 13,-13,-38,-61,-78,-88,-90,-85,-73,-54,-31, -4, 22, 46, 67, 82, 90,} {89, 79,59, 33, 2,-28,-56,-76,-88,-90,-81,-62,-37,-7, 24, 52, 74, 87, 90, 83,66, 41, 11,-20, -48,-71,-86,-91,-84,-69,-45,-16, 16, 45, 69, 84, 91, 86,71, 48, 20,-11,-41,-66,-83,-90,-87,-74, -52,-24, 7, 37, 62, 81, 90, 88,76, 56, 28, -2,-33,-59,-79,-89,} {89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75, -89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89, 89, 75, 50,18,-18,-50,-75,-89,-89,-75,-50,-18, 18, 50, 75, 89,} {88, 71, 41,2,-37,-69,-87,-89,-74,-45, -7, 33, 66, 86, 90, 76, 48,11,-28,-62,-84,-90,-79,-52, -16, 24, 59, 83, 91, 81, 56,20,-20,-56,-81,-91,-83,-59,-24, 16, 52, 79, 90, 84, 62, 28,-11,-48,-76,-90,-86,-66,-33, 7, 45, 74, 89, 87, 69, 37, -2,-41,-71,-88,} {88,67, 31,-13,-54,-82,-90,-78,-46, -4, 38, 73, 90, 85, 61,22,-22,-61,-85,-90,-73,-38, 4, 46, 78, 90, 82, 54,13,-31,-67,-88,-88,-67,-31, 13, 54, 82, 90, 78, 46,4,-38,-73,-90,-85,-61,-22, 22, 61, 85, 90, 73, 38,-4,-46,-78,-90,-82,-54,-13, 31, 67, 88,} {87, 62,20,-28,-69,-89,-84,-56,-11, 37, 74, 90, 81, 48, 2,-45,-79,-91,-76,-41,7, 52, 83, 90, 71, 33,-16,-59,-86,-88,-66,-24, 24, 66, 88, 86, 59,16,-33,-71,-90,-83,-52, -7, 41, 76, 91, 79, 45, -2,-48,-81,-90,-74,-37,11, 56, 84, 89, 69, 28,-20,-62,-87,} {87, 57, 9,-43,-80,-90,-70,-25, 25,70, 90, 80, 43,-9,-57,-87,-87,-57, -9, 43, 80, 90, 70, 25,-25,-70,-90,-80,-43, 9, 57, 87, 87, 57, 9,-43,-80,-90,-70,-25, 25, 70,90, 80, 43, -9,-57,-87,-87, -57,-9, 43, 80, 90, 70,25,-25,-70,-90,-80,-43, 9, 57, 87,} {86, 52, -2,-56,-87,-84,-48, 7, 59,88, 83, 45,-11,-62,-89,-81,-41, 16, 66, 90, 79, 37,-20,-69, -90,-76,-33,24, 71, 91, 74, 28,-28,-74,-91,-71,-24, 33, 76, 90, 69,20,-37,-79,-90,-66,-16, 41, 81, 89, 62, 11,-45,-83,-88,-59, -7, 48, 84,87, 56, 2,-52,-86,} {85, 46,-13,-67,-90,-73,-22, 38, 82, 88, 54,-4,-61,-90,-78,-31, 31, 78, 90, 61, 4,-54,-88,-82, -38, 22, 73, 90, 67,13,-46,-85,-85,-46, 13, 67, 90, 73, 22,-38,-82,-88,-54, 4, 61, 90, 78,31, -31,-78,-90,-61, -4, 54, 88, 82, 38,-22,-73,-90,-67,-13, 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